Why is math proof so important in scientific research?

[Almagor]

Many things that were once thought of as self evident (such as that the Earth is flat) were later shown to be not true. Doesn't the fact that what is now viewed as self evident may later be shown to be not true indicate that Mathematics can not "prove" anything? Wasn't there a famous mathematician in the 1960's that "proved" mathematically that nothing can be proved by mathematics? Lastly, isn't it too convenient to be able to "define" something to be true without having to prove it some way. I thank anyone in advance that has an answer to these questions.

 

[DaveC426913]

Many things that were once thought of as self evident (such as that the Earth is flat) were later shown to be not true.

It was once thought to be self-evident that it was self-evident that the Earth was flat. It was later shown to not be true that it was later shown not to be true.

i.e. it is a myth that we used to think Earth was flat. Peasants may have; no educated person did.

Doesn't the fact that what is now viewed as self evident may later be shown to be not true indicate that Mathematics can not "prove" anything?

Math is based on axioms.

IF we define 1+1=2 (axiom)
THEN we can prove that 1+2=3.

 

[Almagor]

Thanks for your reply. Yes, but if I define myself as a bird, I may be able to "prove" that I can fly, but what good is that, I can't fly. If I define 1+1 =3 (my axiom) then you can't prove 1+2=3. If you can't prove your axiom but merely define it, how is your definition any better than mine? If the word "prove" does not mean that something is true, what good is the word? Please, I am not trying to offend but I think my points are valid.

 

[DaveC426913]

Thanks for your reply. Yes, but if I define myself as a bird, I may be able to "prove" that I can fly, but what good is that, I can't fly.

The entire rational world is built on some assumptions we must make. Here are a few:
A line of length n will remain of length n one each time we measure it.
I think therefore I am.
A+B=B+A
1+1=2.

These are all things we cannot "actually" prove. They are things we must assume if we are to build a rational world. (Rational meaning it has conceivable rules we can count on).

If I define 1+1 =3 (my axiom) then you can't prove 1+2=3. If you can't prove your axiom but merely define it, how is your definition any better than mine?

There is aboslutely nothing wrong with your axiom; it is every bit as valid. The question is: how useful is it in describing the real world? We use the axiom 1+1=2 because, when we apply it to the world we see around us, it results in logical and consistent answers, allowing to to build further without running into contradictions (such as: we have enough fence to go around the circumference of our circular corral instead of discovering too late that we only have enough to surround it if it is hexagonally-shaped).

 

[Almagor]

Dear Dave, When you apply 1+1=2 to the real world we see around us and find that it works, isn't that the ultimate proof that 1+1=2 is true and not merely an axiom? Thank you for taking the time to answer my questions. You have been very helpful and I hope we communicate again some time. Just as aside, The statement, " I think therefor I am." is a famous statement but is incorrect. I trained in Raja and Jnana Yoga in an ashram in the 60's. We did many mental exercises that proved to me through personal experience that "I am" whether I think or not. In fact, I can realize that "I am"
more fully when my mind is completely silent.

Thanks again.

 

[HallsofIvy]

you know the funny thing is, the very principle axiomatic foundations of maths is based upon "self-evedent truths" i.e. we can't prove it is, we just accept that it is, i.e. there is no rationality or logic to rationality or logic itself;

You are simply mistaken to think that math is based on "self-evident truths". "Axioms" and "postulates" are NOT "self-evident"- they are statements that are assumed to be true for the purpose of argument. All mathematics says "IF these are true, then ...".

I looked back and found that I had said the same thing in this thread two and half years ago!

 

[HallsofIvy]

Since you had restarted this thread after it had had a well deserved rest for two and half years,

Many things that were once thought of as self evident (such as that the Earth is flat) were later shown to be not true. Doesn't the fact that what is now viewed as self evident may later be shown to be not true indicate that Mathematics can not "prove" anything?

I, and others, said two and half years ago, that thinking that axioms are "self evident" is an error.

Wasn't there a famous mathematician in the 1960's that "proved" mathematically that nothing can be proved by mathematics?

No, there wasn't. You may be thinking of Goedel's proof. But that was in the 1920's and what he proved was that any system of axioms (large enough to encompass the natural numbers) is either "incomplete" (there exist statements you can neither prove nor disprove) or "inconsistent" (you can prove both a statement and its negation). This does not mean that nothing can be proved. There exist good reason to believe that all axiom systems we use are consistent so that just says that there will be some things we cannot prove without adding new axioms. No problem with that.

Lastly, isn't it too convenient to be able to "define" something to be true without having to prove it some way. I thank anyone in advance that has an answer to these questions.

It would be if we stopped there. But we don't. We "define" Euclidean geometry to be a system in which the parallel postulate is true. If we stopped there and refused to consider any other kind of geometry, we would be wrong (or at least limited). But we don't. We also "define" other kinds of geometries in which the parallel postulate is false and see what happens in those. That is what I meant about "if... then...". If "given a line and a point not on that line, there exist a unique line parallel to the given line through the given point" then all the results of Euclidean geometry. But also if "given a line and a point not on that line, there exist more than one line parallel to the given line through the given point" then all the results of hyperbolic geometry and if " given a line and a point not on that line, there exist no line parallel to the given line through the given point" then all the results of elliptic geometry.

While mathematics "assumes" things for one kind of mathematics, in total, it considers all possiblilties.

If you want to say that mathematics alone cannot "prove" statements about nature or, say, physics, then you would be perfectly correct. That is not what mathematical proofs do. I will say the same thing I said above (and two and a half years ago!)- all statements in mathematics are of the form "If ... then ...". All mathematics does is say "If" the axioms are true, then these are the things that will follow. You cannot argue that mathematics is "wrong" by arguing against the axioms, though you could argue that it is useless because we do not know whether those things are true or not. But history has shown that, indeed, mathematics is very useful! And it is useful in so many different ways specifically because it "assumes" so many different things. For any application, there is bound to be some form of mathematics that "assumes" just what you want!

 

[DaveC426913]

Just as aside, The statement, " I think therefor I am." is a famous statement but is incorrect.

Perhaps you should read up on the meaning of the phrase before deciding it is incorrect.


Descartes decided to see what would happen if he doubted everything. He quickly digressed to doubting his own existence.

He concluded that, in order to doubt his own existence, there had to be something doing the doubting. Whatever that something is, it defines I.



I'll phrase the concept within your learnings:

"I am" whether I think or not.

There is something doing the refraining from thinking. That something is I.

 

[HallsofIvy]

Though it would be unfortunate to define oneself as "that which does not think"!

 

[HallsofIvy]

It is not what I consider is, all I am saying is, how is it possible to "prove" (i.e. that which requires logic/rationale) axioms themselves, if logic and rational contain the very axioms we are trying to prove?

It is out of that manner of thinking that I said (and as another poster has said, it is merely "defined") that it is "self evidently true" (i.e not proven but defined)

You are apprently not understanding anything being said here. Have you actually read and thought about it all? We have said repeatedly that axioms are not "self evident" yet you continue to use that phrase. The whole point of mathematics is to say "if this is true, then what are the consequences?"

Do you see no value in saying to yourself "If I do this, what will be the consequences"?

 

[DaveC426913]

You are apprently not understanding anything being said here. Have you actually read and thought about it all? We have said repeatedly that axioms are not "self evident" yet you continue to use that phrase. The whole point of mathematics is to say "if this is true, then what are the consequences?"

And note that axioms define the very essence of the scientific method: we always move forward with the proviso that we may be wrong about anything and everything. (Though that does not slow our forward progress.)