Thread: proof?

Originally Posted by Licity

You cannot prove it at all, because there are an infinite number of possible squares. Squaring a side will always get the area of the square, but there is no way to empirically prove this.

my whole point is that you cannot empirically prove that statement. You can mathematically prove it. For this case, it is generally actually taken as an axiom that by definition the square with sides of unit length has unit area.

then a square with sides of lenth l can be broken down into l^n squares of unit area so that square has area of l^n because we just measure the area by adding up the area of the unit squares.

That might be a little unsatisfying so I started a new thread here.

This is the fundamental difference between math and science. It is true that trial and error play a large role in arriving at hypothesis but stating formulas and calculating their values on given inputs is not mathematics. Mathematics possesses the certainty that many people try to ascribe to science.

Is maths a science? It really depends on the definition, but even if it is not a science, it shares many characteristics with it. But obviously using logic to manipulate numbers (an abstract concept that does not require observation), it's possible to completely prove something.

Whereas in what you might call 'conventional science', completely proving something would mean making an infinite amount of observations, which clearly is not practical.

I know this has already been tackled, but I'll say it anyway.

Really, when most people of a scientific mind say something has been 'proven' (outside of any situation involving pure logic), all they are really saying is "there's a lot of evidence for this particular explanation and it hasn't been falsified yet", but obviously saying that is a bit of a mouthful. On a day-to-day basis outside the philosophy classroom, we tend to round [insert very high degree percentage of certainty here] up to 100% for practical reasons.

What's really interesting is that a lot of people of a religious mindset use the same practical form of proof as science in day to day life, and indeed accept scientific evidence (for example in the justice system), and only say a bad word about it when it conflicts with their beliefs. You tell them that their partner is cheating on them, or X murdered Y, and they'll probably demand evidence. It's only when religion comes in that this goes out the window.

Originally Posted by aaronasterling

Technically, it consists of theorems and proofs. The capacity for proof is what separates mathematics from science and is of course the step that licity left out in his argument.

Not really. What I mean about hypotheses and theses is that, if the hypothesis is met, then the thesis can be assumed as true, because the hypothesis implies the thesis. Mathematics is the study of what things imply what others.

I'm not sure what scientific validity is supposed to mean other a statement that is falsifiable and not yet falsified.

"Falsifiable" is a very dangerous adjective. What is meant by "falsifiability" in the scientific method is that a potentially wrong model or truth should be falsifiable. Mathematics doesn't break that rule, because it's all made up of definitions and immediate implications. Nothing to be potentially unproven. Of course, the mathematician may make mistakes, but that is another thing. There is no limit of known facts in mathematics as there is in science, and that's why mathematics is so cohesive.

Originally Posted by aaronasterling

It was a bit harsh to say that they would die laughing. One as arrogant as me would no doubt have a good chuckle.

When you have tested a 1000 squares, how do you know that it will work for the next one?
You prove it

You don't really "prove it". There is a certain logical method called induction. It consists of analysing a behaviour through iterations, observing its tendency, and concluding some truth is valid for many situations, without having to test each of them.

This is the fundamental difference between math and science. It is true that trial and error play a large role in arriving at hypothesis but stating formulas and calculating their values on given inputs is not mathematics. Mathematics possesses the certainty that many people try to ascribe to science.

The reason mathematics is never wrong (assuming no mistakes in the logical reasoning) is because mathematics doesn't have a limit of known facts, like physics and chemistry do. There is no sine qua non knowledge to mathematics. Mathematics is made of definitions and immediate implications. Science is made of observed facts, definitions and immediate implications.

Hey Kromoh, glad you still want to play.

Originally Posted by Kromoh

Not really. What I mean about hypotheses and theses is that, if the hypothesis is met, then the thesis can be assumed as true, because the hypothesis implies the thesis. Mathematics is the study of what things imply what others.

correct. It was a misunderstanding on my part. A lot of mathematicians would refer to what you call a hypotheses as a 'condition'. For example the conditions of Sylow's theorem which I gave you earlier are that:

1) G is a finite group
2) the order of G (often written |G| which just refers to how many members it has) is divisible by p^n but not p^(n+1) where p is a prime

so yes, you are correct. misunderstanding of definitions.

Originally Posted by Kromoh

You don't really "prove it". There is a certain logical method called induction. It consists of analysing a behaviour through iterations, observing its tendency, and concluding some truth is valid for many situations, without having to test each of them.

You do prove it. I did in this tread. You do not need to perform one test. as you are using it, induction is not a logical method. What you are talking about is physical induction and is one of the foundations of the validity of science. An example of logical induction (a method of proof in mathematics) goes as follows:

Theorem: the sum of all integers from 1 to n is n(n + 1)/2

proof:
we have 1 = 1(1 + 1)/2 = 1 so the assertion is true for 1.

now suppose that the assertion is true m - 1. We have

(m - 1)m/2 + m = (m^2 - m)/2 + 2m/2

= (m^2 + m)/2

= m(m + 1)/2

so the assertion is true.

The method of induction itself can be proven but I'm not going to do that unless requested. I'd have to hit the books for that . The idea is that if it's true for 1 and it's true that when true for any integer m, it's true m + 1, then it is true for all integers > 1.

Originally Posted by Kromoh

The reason mathematics is never wrong (assuming no mistakes in the logical reasoning) is because mathematics doesn't have a limit of known facts, like physics and chemistry do.

again, it's because we prove it. It has nothing to do with the cardinality of available facts.

again though, this boils down to a definition of science.

The whole point of science is that it limits the available mechanism for declaring a statement valid by imposing the scientific method. Mathematics, in employing proof, moves outside that limit.

Theorem: the sum of all integers from 1 to n is n(n + 1)/2

proof:
we have 1 = 1(1 + 1)/2 = 1 so the assertion is true for 1.

now suppose that the theory is true for all m - 1. We have

(m - 1)m/2 + m = (m^2 - m)/2 + 2m/2

= (m^2 + m)/2

= m(m + 1)/2

so the assertion is true.

This seems a weird way of doing it... I'm not sure if it's right.

Don't you have to show that it's true for k = 1, and then that the truth of P(k) implies the truth of P(k+1)..? I don't understand how k-1 is involved?

I would do it as,

P(n) <=> sum of integers from 1 to n = 1/2*n*(n+1)
P(1) <=> sum of 1 = 1/2*1*2 = 1, TRUE
P(k) <=> sum of integers from 1 to k = 1/2*k*(k+1)
=> sum of integers from 1 to k+1
= sum of integers from 1 to k + k+1
= 1/2*k*(k+1) + k+1
= (k+1)(1/2*k +1)
= 1/2*(k+1)(k+2)
= 1/2*(k+1)((k+1)+1)
<=> P(k+1) TRUE

As P(1) is TRUE and the truth of P(k) implies the truth of P(k+1), P(n) is true for all n E N.

By the way, I don't think you're right in saying that the inductive principle can be proved... I think it's axiomatic?

Originally Posted by Xei

This seems a weird way of doing it... I'm not sure if it's right.

Don't you have to show that it's true for k = 1, and then that the truth of P(k) implies the truth of P(k+1)..? I don't understand how k-1 is involved?

it's right except for the statement:
"now suppose that the theory is true for all m - 1"

this should read "suppose the statement is true for m - 1"

I'll edit my above proof to reflect this. thanks for pointing it out.

the difference between the m - 1, m formulation and the m, m + 1 formulation is just a matter of taste and convience. Sometimes, one or the other will lead to easier calculations. They both serve the same purpose for the inductive step though as both allow you to go from n to n + 1.

Originally Posted by Xei

By the way, I don't think you're right in saying that the inductive principle can be proved... I think it's axiomatic?

You have a choice of axioms. You can go from the statement that "any set of natural numbers has a least element" to the theorem of induction or you can go from the axiom of induction to the statement that any set of natural numbers has a least element.

Heres the proof from Undergraduate Analysis by Serge Lang.

we take the property of well-ordering as an axiom of the natural numbers. It says that "Every non-empty set of natural numbers has a least element."

Suppose that for each positive integer we are given an assertion A(n), and that we can prove the following two properties:

(1) The assertion A(1) is true.
(2) For each positive integer n, if A(n) is true, then A(n + 1) is true.

Then for all positive integers n, the assertion A(n) is true.

proof: Let S be the set of all positive integers n for which the assertion A(n) is false. We wish to prove that S is empty, i.e that there is no element in S. Suppose there is some element in S. By well-ordering, there exists a least element, n_0 in S. By assumption, n_0 != 1, and hence n_0 > 1. Since n_0 is least it follows that n_0 - 1 is not in S, in other words the assertion A(n_0 - 1) is true. But then by property (2), we conclude that A(n_0) is also true because n_0 = (n_0 - 1) + 1. This is a contradiction, which proves what we wanted.

Or we can do:


Suppose that for each positive integer we are given an assertion A(n), and that we can prove the following two properties:

(1) The assertion A(1) is true.
(2) For each positive integer n, if A(n - 1) is true, then A(n) is true.

Then for all positive integers n, the assertion A(n) is true.

proof: Let S be the set of all positive integers n for which the assertion A(n) is false. We wish to prove that S is empty, i.e that there is no element in S. Suppose there is some element in S. By well-ordering, there exists a least element, n_0 in S. By assumption, n_0 != 1, and hence n_0 > 1. Since n_0 is least it follows that n_0 - 1 is not in S, in other words the assertion A(n_0 - 1) is true. But then by property (2), we conclude that A(n_0) is also true. This is a contradiction, which proves what we wanted.

Only thing I want to refute:

Originally Posted by aaronasterling

The whole point of science is that it limits the available mechanism for declaring a statement valid by imposing the scientific method. Mathematics, in employing proof, moves outside that limit.

What I meant on my last post is that science should be "falsifiable", which is the only thing that mathematics isn't. The reason to this is because sciences are limited by the ability and technology available to observation. When human beings didn't know radioactivity, Thomson's atomic model was fine and enough. But it wasn't completely correct, because he didn't have the tools to observe subatomic particles at the time. Mathematics doesn't have such limits - you could sit down one day and deduce the entire field of mathematics all alone (of course you'd have to be one little genius, but it's possible by definition).

The issue with falsifiability cleared, mathematics fits all other parts of the scientific method (in many cases better than other sciences) - so yes, mathematics can be considered a science. All mathematical conclusions, implications, deductions and inductions abide by the scientific method.

I still fucked up my proof!!!!! :

"suppose the statement is true for all m - 1"

should be suppose the statement is true for m - 1. The problem is that the all is confusing and I don't know why i want to keep putting it in there. much better. I'll edit it again