Why is math proof so important in scientific research?

[matt grime]

Instead of this pointless philosophical musing, why not say why you consider the parallel postulate to be 'self evidently true', Anhar, and then explain hyperbolic or spherical geometry?

 

[Anhar Miah]

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)

 

[CompuChip]

I personally like this example of "pure" logic.

Surely, you will agree with a reasoning like this:

1. All humans are mortal.
2. You are a human.
3. Therefore, you are mortal.

You could call this a proof of the statement "you are mortal" with the first two items as axioms. If you accept the axioms, you cannot refute the conclusion.

Now consider this argument:

1. All humans have two heads.
2. You are a human.
3. Therefore, you have two heads.

The argument is precisely the same, and therefore the proof is equally valid. OF course, your intuition tells you that something is wrong here, because obviously the first premise is not true. But if you accept the first two items, again the third must follow. It is still a good proof, although it doesn't describe our real world. But then again, that is not the purpose of mathematics in general (in physics, however, we would reject the second proof because the axioms used there do not correspond to reality). Mathematics is , basically, just concerned with the question: if we accept these axioms, what can we deduce from them. Physics is more concerned with: are these axioms physically acceptable (e.g. relating to the real world) and then applies mathematics to draw conclusions about the world.

 

[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?

Well, you have already been told, repeatedly that it is NOT possible to prove axioms themselves so your basic question makes no sense.

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)

Then it is a language problem. "Defined to be true" is not at all the same as "self evidently true".

 

[slider142]

The language is defined through axioms, which, at most, define undefined objects only by their behavior; that is to say, their relationship to other undefined objects. Ie., points, lines, etc. are the undefined objects of Euclidean geometry. The axioms are given truth values; whether those truth values agree with some real world object or not is not relevant; only that they be consistent (non-contradictory), as a contradiction is how we change truth values. There is more to this than simple logical machinery of black boxes, though, as is evidenced by the existence of undecidable propositions.

 

[CRGreathouse]

The argument from pragmatics.

p1. Technology makes people happy.
p2. Technology is born from rational thought.
p3. It is rational to like that which leads to us being happy.

Therefore,

c1. Rational thought leads to us being happy.
c2. Rational thought is rational.

p4. "φ is born from ψ" implies "ψ leads to φ".
p5. "ψ makes χ" and "φ leads to ψ" implies "φ leads to χ"
p6. What makes people happy makes us happy.
p7. If it is rational to like something, it is rational to do that thing.

c1. Rational thought leads to technology (p4, p2)
c2. Rational thought leads to happy people (p5, p1, c1)
c3. Rational thought leads to making us happy. (p6, c2)
c4. It is rational to like rational thought. (p3, c3)
c5. It is rational to be rational. (p7, c4)


So the fairly uncontroversial c5 can be proven from a stack of highly questionable premises. p1, "technology makes people happy", has already been attacked on the grounds that some technologies (say, mustard gas) don't make people happy. I'll instead question the quantifier in conjunction with p6. Just because technology makes someone happy doesn't mean that it should make me happy. p7, one of my 'technical axioms' that you used in going from your c1 to c2, also seems questionable. Many more people like football than actually play.

 

[CRGreathouse]

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)

I've decided to think of axioms and theorems as entirely different classes of objects. In particular, I now consider the statement of each axiom as a theorem with a one-line proof. In Peano arithmetic:

Theorem: 0 is a number.
Proof:
1. 0 is a number, QED.

So the axioms themselves have no meaning (to me) as such; they're just used to prove things. From this perspective it's evident that axioms are needed -- you can't 'make a move' without the axioms. Of course here I include the logical axioms as well.

 

[Anhar Miah]

Well, you have already been told, repeatedly that it is NOT possible to prove axioms themselves so your basic question makes no sense.

:) This is extremely comical, that’s the exact point I've been trying to point out (via the questions), and how can you (or others) be arguing against me when we are both agreeing on the same point? :D

Then it is a language problem. "Defined to be true" is not at all the same as "self evidently true".

Fair enough, my words have been misinterpreted to mean (by some); that other than which I intended.

And for the record I grew up in the UK with everyone speaking english around me, therefore in that sense I may conclude that; that makes a me a "Native Speaker" :D

 

[CompuChip]

:) This is extremely comical, that’s the exact point I've been trying to point out (via the questions), and how can you (or others) be arguing against me when we are both agreeing on the same point? :D

So may we then conclude that either your question has been answered, or you still haven't managed to clearly explain the point.

And for the record I grew up in the UK with everyone speaking english around me, therefore in that sense I may conclude that; that makes a me a "Native Speaker" :D

Even native speakers have language problems sometimes. Not because they speak the language badly, but because they didn't express themselves carefully enough. Especially in mathematics being concise and to-the-point is very important.

 

[matt grime]

:) This is extremely comical, that’s the exact point I've been trying to point out (via the questions)

No it wasn't. Your point was that axioms are 'self evidently true'. Which is entirely different. No one else has misrepresented you - you have not explained yourself properly, and made semantic/linguistic errors. Your misuse of a semi-colon (at least twice in post 38) indicates that you're not above such things, irrespective of where you grew up.

 

[DeaconJohn]

so first off i will state that math is a human language... whenever an article is published or a paper written and it is said to be true it must be sited for where the info came from and often times the source has to be checked as well because, who says they are right anyway. when a scientist finds mathematical proof that something is correct and others look over the work and see there are no mistakes it becomes accepted as proof and no sources are required... why does math always prove something correct? anybody got some good answers?

So, Shamrock. I was wondering why there was a line through your name. But then i noticed that Computchip asked and Dave answered that that is because you have been banned from the site, "at least temporarily."

Is that true? Is that the reason that there is a line through your name?

If so, what was the sequence of events that ended-up with you being banned?

I'm a newbee. The reason I'm asking is I don't want any lines through my name if I can possibly avoid it. I've read the policy, but "real stories" are always useful to aid in one's understanding.

So, if you could be so kind as to share that information with us, what happened?

DJ

 

[CRGreathouse]

Even native speakers have language problems sometimes. Not because they speak the language badly, but because they didn't express themselves carefully enough. Especially in mathematics being concise and to-the-point is very important.

Very important, but also quite hard. I see it as one of the most valuable skills in mathematics -- both for mathematics itself and for applications to other fields.

 

[Anhar Miah]

No it wasn't.

well that sir is plainly subjective, what one may find satire another may not.

Just out of curiosity how many languages do you speak?

Hmm, you seem to have mistaken this for a philosophy forum.

In the same essence :

"Hmm, you seem to have mistaken this for an English language forum."

Now just relax before you get all wound up! I'm joking with you :)

I'm sure you maths guys are not that robotic, are you?

 

[CRGreathouse]

Robotic? No. But matt grime does have a tendency to ruthlessly mark things he doesn't want to talk about as philosophy. (No hard feelings, Matt!)

In fairness, musing on the purpose of logic and rationality is metaphysics, a branch of philosophy.

 

[matt grime]

I'm happy to talk philosophical matters. I have a copy of Wittgenstein staring me in the face as I write this. I just don't pretend that it's mathematics. There is a philosophy subforum if you want to discuss it.

On the subject of language - since this you're writing about the meaning of terms it helps more than normal if you use the terms correctly, otherwise misunderstandings and miscommunications are bound to occur. It doesn't matter if English is your first or tenth language.

 

[Gib Z]

I think the main point one needs to understand is that an axiom is the proposed truth, and then theorems from that axiom are just things that are true if that axiom holds- no one ever says anything about that axiom being true or not. Say the axiom was that one Apple is cheaper than one Orange. Within this system I could then say 2 apples are cheaper than 2 oranges. Our axiom may not be true at all - but IF it is, then the next statement follows.

 

[Anhar Miah]

Matt I totally agree with you, one needs to exercise “correct terminology”, I’m not a mathematician like yourself, so I’m not as rigorous and precise in terms of using mathematically orientated terminology/ descriptions.

P.S. you too made a grammatical error :) please check your last post again!

 

[CRGreathouse]

I'm happy to talk philosophical matters. I have a copy of Wittgenstein staring me in the face as I write this.

Fair enough, I cede the point; I was only teasing.

 

[matt grime]

s

well that sir is plainly subjective, what one may find satire another may not.

I wasn't referring to the comedy of the situation, but to your belief that what Halls's point was the same as yours. Anyway, I hope that we've convinced you that axioms are not "self evidently true". Indeed, if one attempts to reason in an inconsistent system then one will be able to show that the axioms are theorems of the system as are their negations.

 

[Anhar Miah]

I wasn't referring to the comedy of the situation

I could apply your own strict criterion of specification; from your statement it is ambiguous:

No one else has misrepresented you - you have not explained yourself properly.

But relax Matt The English language is one of the most imprecise and ambiguous language.

Now I’m more than happy to say, yes my terms were incorrect, but the underlying point was the same as yours.

The Use of “Self Evidently true”, is not the same as “Defined” but here is what my internal thinking was:

Evidently from Evidence as in “from Proofs”

Self as in the “self referenced” sense, i.e. without any external body,

True as in a “statement of truth”

Thus “Self Evidently” is a proof based upon self reference, a self which is defined by us, in essence anything that is proved by self reference is “proof less” thus we have an axiom or a self that is defined but without proof and can be said to be true (by virtual of definition) or statement of truth .

But I concede, it would be much easier just to say “defined” :)

 

[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"?