Why is math proof so important in scientific research?

[shamrock5585]

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?

 

[CompuChip]

You mean: why do people accept a theorem from a reference once it is generally agreed upon that the proof of that theorem there is correct?

 

[shamrock5585]

well yeah... i mean if someone comes up with a theorem... others check for errors and if the math is correct... it is generally accepted... with most arguements you have to argue sources and language terminology... but if you prove something mathematically it is pretty much correct by default.

 

[DaveC426913]

Math proofs are derived from a very very small set of fundamental axioms. Fundamental, like 1+1=2 and two straight line segments in Cartesian space intersect at, at most, one point. There's not a lot ofpoint in taking up any science unless you grant these first few axoims.

From those first few, all others can be directly created. A new, published math proof consists of showing that your new formula can be derived directly from existing axioms. No math proof is ever accepted until many colleagues have pored over it and concluded that there is no flaw.

In a nutshell, if one accepts the initial underlying axioms, and the rest of the math is done correctly, one has no choice but to accept the new formula.

 

[DaveC426913]

well yeah... i mean if someone comes up with a theorem... others check for errors and if the math is correct... it is generally accepted... with most arguements you have to argue sources and language terminology... but if you prove something mathematically it is pretty much correct by default.

You seem to know the answer to your own question. Are you just having fun?

 

[shamrock5585]

basically just looking for a good explanation of WHY... thanks

 

[CompuChip]

So to summarize, I guess the answer to "why do we accept a theorem once it is proven?" would be: "by definition of a proof".

 

[maze]

so first off i will state that math is a human language...

What? No. The symbols used in math could, in some sense, be considered a language, but they are not math itself.

If I make up my own symbols but solve the same problem, I would not be doing "different math", I would just be writing it a different way.

 

[DaveC426913]

What? No. The symbols used in math could, in some sense, be considered a language, but they are not math itself.

If I make up my own symbols but solve the same problem, I would not be doing "different math", I would just be writing it a different way.

I believe his point is more like 'math does not exist without humans'.

 

[shamrock5585]

yes but if i make up my own symbols for english and speak them when i see the symbols I am still speaking english just not writing english...

 

[shamrock5585]

So to summarize, I guess the answer to "why do we accept a theorem once it is proven?" would be: "by definition of a proof".

well i guess... but what i was getting at is that if you are arguing a point you need to site your sources and prove their credibility but in math you just need to prove that YOU didnt make any mistakes.

 

[matt grime]

I think you're under a slight misapprehension. In mathematics it is very important to cite (not 'site') any sources you use - you're never going to prove everything from first principles. Of course many things are sufficiently self evident, or well known, as to require no citation - you wouldn't bother citing Fermat if you invoke Fermat's Little Theorem, for example, nor would you even name Lagrange if you said "since ord(x) divides |G|" these days.

 

[CompuChip]

well i guess... but what i was getting at is that if you are arguing a point you need to site your sources and prove their credibility but in math you just need to prove that YOU didnt make any mistakes.

Yes, in math you are trying to prove that YOU didn't make any mistakes. If you did use work from someone else (which you have to cite, of course!) you already assume that they are correct (because the steps in them were already checked or assumed correct from cited works, etc., all the way back to first principles.)

Why is there a bar through the name?[/size]

 

[DaveC426913]

Why is there a bar through the name?[/size]

That means he's been banned, at least temporarily.

 

[Anhar Miah]

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;

Is logic and rationality ultimately purposeless?

even that is even more twisted, imagine if you will, that you are asked to make a purposeless machine, can you make it?

(a) Well if you make the machine, even if it does nothing, it was made with a purpose (i.e. to be purposeless) but that then makes it null, becuase in creating purposeless requires a purpose in the first instance

(b) you don't create any machine at all, but then you contradict the initial point, i.e. you can't make it, but at the same time you "can't; can't make it"

eek! :D

 

[DaveC426913]

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;

Well now, I don't know if it follows that it's not rational.

Some of these axioms we actually define to be so.

eg. 1+1=2 (and only 2). While we can't "prove" it, we have defined this to be so.
Same with others.

 

[Anhar Miah]

we have defined this to be so.

Thats what's funny about it!

 

[CompuChip]

Yes, we build math from axioms.
You can take that as an axiom, if you want. I also don't know whether statements like "there is no rationality or logic to rationality or logic itself" are really relevant in this thread. I could argue about making assumptions and using intuition as a basis for science, but I won't :)

 

[Littlepig]

why? because the definition: We assume something, and we start from that assumption(like 1+1=2), and than we make this: if 1+1=2 than 2+1=1+1+1 so, 1+1+1=2+1==3.(and I've defined the 3 number with the premiss 1+1=2). I've just made a definition that, assuming 1+1=2, than 1+1+1=3. Can you disproof my proof? No, because is just a definition.

However, from this definition, one can proof that exists infinite natural numbers, by a logical method called induction. Now, if you imagine that induction is like defining that pi=perimeter/diameter than you can't deny my proof of infinite natural numbers, because that would tell you that: pi≠perimeter/diameter or 1+1≠2. And that is contradicting your definitions. So, It must be true that exists infinite natural numbers.
Now, the whole math is based in this kind of logical concept. Even the most complex idea like derivative, integral, limit, is based on basic axioms like that one, so, it is true because the assumptions you used were all derived from axioms(definitions), if the theorem isn't true, than some definition is contradicting other or itself.

However, that mean that your definitions aren't always good. Exists basically 2 crisis in mathematical understanding. One was the definition that if you have 2 lines, you can always express the larger one as fraction parts of the other(made by the greeks that lead to the Zeno's Paradox and an re-formulation of great deal of mathematical definitions about real numbers) and the other that lead to the mathematical crisis of 19 century with the Russell's paradox or Burali-Forti paradox. that made some reformulation of great deal of maths definitions with Cauchy among others. Why, because definitions can't be made to contradict then-selfs.

 

[Werg22]

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;

Is logic and rationality ultimately purposeless?

even that is even more twisted, imagine if you will, that you are asked to make a purposeless machine, can you make it?

(a) Well if you make the machine, even if it does nothing, it was made with a purpose (i.e. to be purposeless) but that then makes it null, becuase in creating purposeless requires a purpose in the first instance

(b) you don't create any machine at all, but then you contradict the initial point, i.e. you can't make it, but at the same time you "can't; can't make it"

eek! :D

I would be careful in stating that axioms are self-evident truths. This is only viable when there exists fairly simple physical representations of the deductive system at hand. For example, the axioms of incidence are intuitively self-evident when we represent lines as, say, rays of light (and points as photon small regions of these rays). However, attributing the terms of Euclidean geometry to other physical objects, the relationships described by the axioms could still hold without them being intuitively obvious.

 

[matt grime]

I think it is neither true to say 'axioms are self evident truths' nor helpful to sloganeer that 'logic is irrational'.

We do not take axioms to be self evidently true. The canonical example of why this is the wrong way to think of it is the parallel postulate in geometry. If I accept it I get Euclidean geometry. I can use its negation and get other (more?) interesting geometries too. A similar thing occurs with the axiom of choice.

It is more accurate to say we start with some set of axioms, and see what theorems follow. Whether this produces something useful or interesting will be seen with time.

 

[HallsofIvy]

Notice by the way that every mathematical theorm starts "IF ...". That is, it is a conditional statement. Mathematics only says what follows from specified hypotheses and is not concerned with whether those hypotheses are true of false.

The reason mathematics is so generally applicable is because mathematical statements are "TEMPLATES". Any mathematical theory starts with "undefined terms" and axioms that are "accepted without proof" (more technically "postulated"). Other definitions and theorems are given in terms of those. In order to apply a mathematical theory to a given situation, you must assign meaning to those "undefined terms" in a way that makes the postulates true. Of course, since applications always involve some kind of measurement, which is never perfectly correct, the best we can hope for is that a particular theory will "reasonably accurately" fit the application.

 

[Anhar Miah]

I don't see what the issue here is, here is a simpe related question:

"Prove rationally, that rationality is rational";

Go through it carefully..

Anhar,

 

[matt grime]

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

The issue was your unhelpful, and unmathematical, characterisation of axioms as 'self evidently true'. (Something I'm sure did myself, once, before I learned better.)

 

[Anhar Miah]

Well many apologies if that’s how it has come across, so now that you have learned better how would you put it? :D lighten up people!

 

[matt grime]

See above posts by HallsOfIvy and me. In particular note those axioms which are negated in different theories such as geometry: is the parallel postulate 'self evidently true'?

 

[Crosson]

Prove rationally, that rationality is rational

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.

ὅπερ ἔδει δειξαι.

 

[Anhar Miah]

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.

ὅπερ ἔδει δειξαι.

Well that's a fair argument, but I feel it falls a bit short:

p1. Technology makes people happy

This assumes all technology makes all people happy, what about technologies that does not make people happy, such as missile technology or any weapons tech, these cause great unhappiness, thus p1. has to be limited in scope, to some technology makes some people some of the time (because it is also time dependent, yesterday your dial-up made you happy, nowadays your DSL makes you happy)

 

[LukeD]

Well that's a fair argument, but I feel it falls a bit short:

p1. Technology makes people happy

This assumes all technology makes all people happy, what about technologies that does not make people happy, such as missile technology or any weapons tech, these cause great unhappiness, thus p1. has to be limited in scope, to some technology makes some people some of the time (because it is also time dependent, yesterday your dial-up made you happy, nowadays your DSL makes you happy)

You asked for a proof of nonsense. Did you expect anything other than nonsense in return?

 

[Anhar Miah]

You asked for a proof of nonsense. Did you expect anything other than nonsense in return?

Well I was not expecting sense or nonsense, I can not accurately predict what another individual or individuals reply will be in terms of sense of nonsense, hence the best choice is to opt for is one that is neutral.

 

[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 philosphy 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” :)