Multiple proofs of the same statement / proposition

[achref]

Hello friends ,

I have been thinking about whether a mathematical statement , which is true , can be REALLY proved by different proofs.
I think that if we take multiple proofs of the same statement , some of these proofs are "equivalent" , in the sense that :
- They have the same spirit , but seem to be different (for example on approach is geometric and the other is analytical , but they represent the same essence of proof) . In this case, the only difference is that the language used to one proofs (geometric vs analytical ) is not the same. So they are not intrinsically different proofs.
- They attack the same property / feature of the problem .

You can think of a mathematical statement as a person with different aspects of personality that shaped certain characteristics of his . So , we can say that the person has this characteristic , because of different reasons, and those reasons might be "disjoint" in the sense , they have no interaction between each other , they have no effect on one another , but either one of them can lead to that characteristic being existing in that person.

In the same spirit, two proofs might "attack" different aspects of the personality of that statement true . They should not be equivalent , and just being verbalized in a different approach (like geometric vs analytical ), but they could represent a unique , one of a kind reason , for which that claim is true, independently .

My question whether , if a statement is provable to be true , can have two proofs which are intrinsically different ( i don't know how to define it), or "disjoint" , or that every proof is essentially the same as the other but just another representation of it , at a deeper level , even if it doesn't seem to be one (a first sight). I don't know if this question is in relation with Godel's work or has been treated before.

Please share your ideas about this .

Sincerely

 

[Ishuda]

I would say that the answer depends on just what is meant by intrinsically different. For example, the problem is: Given
2 x + 3 y = 10
1 x + 2 y = 10
what are x and y?

One solution could be, from the second equation,
x = 10 - 2 y
and then from the first equation
20 - 4 y + 3 y = 10
or y = 10 and x = -10. That solution requires no knowledge of linear algebra in regards to matrices and their inverses, etc.

However, the solution where that problem is written a 'solution row vector' times a matrix is equal to a row vector, i.e.
( x y ) A = (10 10)
where A is a 2X2 matrix with rows (2 3) and (1 2) and the solution is given as
(x y) = (10 10) A^-1
does require some knowledge linear algebra.
EDIT: Or more generally
x = c A^-1
when x (the unknowns) and c (the constants or givens) are n-tupal row vectors and A is an nXn matrix.

Are these two approaches intrinsically different? I would say they are in one sense but you might not think they are.

 

[achref]

In this particular example , i think they are. One can see the "equivalence " between using a linear system of equations and its equivalent in the multidimensional case , i.e. using matrices and vectors .
And the manipulations using the matrix may lead to the same manipulations on a smaller level on the equations of the system. So both methods are basically equivalent , and just have different dictionary . So they are more equivalent proofs than different, It is kinda like geometric versus analytical .
And i think in this example, there is only one "kind" of proof, all let's say all proofs are equivalent.

If we take , for example , the fact that the square root of 2 is irrational. There are multiple proofs of it. Some are analytical , others are geometric. It is hard to say whether they are equivalent proofs, in a deeper way. It is not as obvious as in the example you gave , Ishuda. . I'm still thinking about it.

 

[Ishuda]

You mean then that an good understanding of linear algebra is the same, intrinsically, as the good understanding of simple substitution? That a good understanding of topology is the same as a good understanding of (through several levels of 'is the same as') a good understanding of arithmetic. That, of course, is your privilege, but I don't think so.

Your question reminds me, in a way, of my 'collection of delta functions' when talking about a solution to a differential equation and its numerical solution. The solution had several aspects involving asymptotic solutions, variations on isotropic solutions, adiabatic solutions, etc. They could all be inter-related and thus were 'all the same solution' but the concepts were different. Were they 'intrinsically different'? I don't think so [despite which delta function you used] but they did require different approaches, i.e. like the geometric and analytic approaches.

So, I would say that the answer also depends on the level of understanding of the person answering the question.

 

[pka]

In this particular example , i think they are. One can see the "equivalence " between using a linear system of equations and its equivalent in the multidimensional case , i.e. using matrices and vectors .
And the manipulations using the matrix may lead to the same manipulations on a smaller level on the equations of the system. So both methods are basically equivalent , and just have different dictionary . So they are more equivalent proofs than different, It is kinda like geometric versus analytical .
And i think in this example, there is only one "kind" of proof, all let's say all proofs are equivalent.

If we take , for example , the fact that the square root of 2 is irrational. There are multiple proofs of it. Some are analytical , others are geometric. It is hard to say whether they are equivalent proofs, in a deeper way.

First let’s take care of your remark about Gödel’s work. His great theorem states: in any axiom system rich enough to contain basic arithmetic there will arise valid statements that are either provable true or false. There are a huge number of academic myths about the meaning and importance of that theorem. Here is a rather honest exploration of the theorem. JANNA LEVIN is an astrophysicist and writer. She has contributed to an understanding of black holes, the cosmology of extra dimensions, and gravitational waves in the shape of spacetime. She is the author of A Madman Dreams of Turing Machines, which won the PEN/Bingham prize.

Now to the other side of your question. Consider two area of mathematics: axiomatic geometry (synthetic without numbers) and analytic geometry. Those two seemly have a great deal in common: points, lines, planes, parallels, perpendiculars, etc. The theorems seem to also overlap. But do your see that it is virtually impossible of any of the proofs to overlap. Axiomatic proofs are just that, devoid of any metric (number) considerations. Whereas analytic geometry set in a metric space so proof are all about metrics (distances, numbers).
I wish that you could give your own example of overlapping proofs.