# Usign postulates to prove validity of a theorem

#### haki

matt grime said:
Ok, let me put it this way.

When I read your arguments this is what I see.

1. Physics/Engineering is so cool because it lets you model all these things. I like solving problems by putting in numbers to formulae.

2. Mathematics is so bad because its just following rules without any reason why.
You have put things out of context.

What I said is that I prefer formulae over proof techniques beacuse I get a result from the formulae that I can see in real world where doing a proof has no visible results.

I would never have claimed that putting numbers into equations is stimulationg, I am sorry if I have missled you to believe soo, but I do think that putting numbers into equations can give you sometimes results that you can observe in the real world compared to nothing that you get from doing proofs that were already done and are stated as theorems. If something was false it would not be allowed to be a theorem would it???

What I mean by understaning is this. You should know the meaning of what is it that you are trying to achieve. Take for example this:

32 = 2^x

find the value of X, if you are a Math teacher, just for fun, ask somebody to give you the result. If somebody would understand what it is that you are trying to achive, that is you need to find the value of x, one would answer 5 because (2*2*2*2*2 = 16 * 2 = 32). But probably they would want to do things with logarithms. Why? Beacuse they have learned a technique on how you obtain the value of x in such a circumstance and they see the pattern. But that is just knowing, if you would understand you would just say the answer is 5. But you as a mathematician would probably say that is just GUESSING??? You got the solution using arithmetic and not algebra! But hey just doing the logarithms needs no understanding from you. Doing it the way I have show had to do with understanding the problem.

Sorry for giving you more of an calculation example but I do hope you did get my point I was trying to make.

Here one example with definitions.

I was surprised that when I said to a colleage who had the highest grade ever possible in Math, I said jokingly that you can aswell make a recursive definition for faculty, and he said, "Wow, that is interesting, I did not KNOW you can do that.". See if he would understood what was the thing he tried to achive he would have know that you can write

n! = 1*2*...*(n-1)*n as

n! = (n-1)!*n
0! = 1

That is what I had in mind with understanding and not knowing.

I think Mathematics are wonderful, but Mathematics lessons are just a horror. Things that could be explained in plain English are all wrapped in some formalities and proofs. And the focus is on formalites oh, better say the Mathematical rigor, everything must be rigorus. Why? Mathematics are wild west. Take a look at geometry. If I recall there is no one "correct"
axiomatization of geometry!! It varies from textbook to textbook! And then you dare questioning people on the rigor, yet you don't even have your facts straight!

For example, If I were to take a physics lessons in Japan or in South Africa the content would be the same, the explanation is bound to differ from professor to professor. But the 2nd newton law will be the same everywhere. Yet if I were to take geometry lessons there are bound to be some differences in the content. If in some scheme you can take something as an axiom and in another sheme you can take that as a theorem, where is the rigor? If Math is "universal" as in universally the same, how can there be no correct axiomatization of geometry?

You have artificial rigor. How can a mathematician preach about precision yet he is usually late for his lessons. That is just being a hippocrit.

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#### Hurkyl

Staff Emeritus
Gold Member
What I said is that I prefer formulae over proof techniques beacuse I get a result from the formulae that I can see in real world where doing a proof has no visible results.
Where do you think formulae come from? There isn't a formula fairy that visits people in the middle of the night to share the secrets of the universe -- people have to come up with them using the proof techniques you so despise.

Sure, some are rather trivial (deriving F = ma from F = dp / dt).
Sure, some are experimental (deriving F = dp/dt).
But others are mathematical -- of the more "concrete" variety such as the Fourier transform, and the "abstract" variety such as the error correcting codes you use to transmit signals.
(I use quotes because concrete and abstract are rather subjective things)

Theorems don't just give us formulas, though -- they tell us about things. For example, they tell us that given a channel for transmitting information, no matter how clever you are, there's an upper limit on how fast you can transmit information.

I find it hard to imagine a thing of infinite precision.
Even from a purely physical perspective... of course there's infinite precision. All of our physical theories are based on there being an "infinite precision" universe out there. E.G. classicaly, a particle has an "infinitely precise" position, whether or not we're capable of measuring it. Quantum mechanically, a state is "infinitely precise", even if we can't hope to know a fraction of the information contained in it.

2 * sqrt(3) is for me of finite precision, 2 is of 1 significat digit precision, and 3 is aswell of 1 SDP, soo the result should be at most of 1 SDP, soo for this to make sense you should write 2.0 * sqrt(3.0), now that is all of 2 SDP.

See I even see numbers diferently, A mathematician would see 1 as 1.0000... or an alternative version 0.999999.... but I see it as a 1 SDP number not as a infinite precision number.
This is a good example of "knowing without understanding" -- you obviously know the rules for manipulating sig figs.

But do you really understand them? You apparently attempt to apply them even when you shouldn't (e.g. when you do have "infinite precision").

And besides, they aren't even a good way to do error analysis -- they are just a quick and easy approximation that usually lets you avoid getting things very wrong. They teach you sig figs because they don't want to teach you error analysis.
(Yes, I was somewhat irritated when I discovered this. I would have been much happier if I was told up front!)

"You apparently attempt to apply them even when you shouldn't"

This, actually, is one of the things I find very reassuring about mathematics. A mathematical theorem tells you when you can use the theorem. It might even tell you exactly if, when, and how it can fail when you can't use it. When teaching, a nontrival amount of time is often spent giving examples of situations in which you cannot apply the theorem, and showing why it fails... and in testing the student to make sure he recognizes when he can and cannot apply the theorem.

(And don't forget that things like "lim (A + B) = (lim A) + (lim B) (when the R.H.S. exists)" are theorems, and not mere "calculations")

But, at least in intro science courses, there is little to no emphasis on when the things you're learning are applicable, what simplifications are being assumed, whether you're learning an approximate or an exact result... *sigh*

Take a look at geometry. If I recall there is no one "correct"
axiomatization of geometry!!
You're right -- the exact same thing can usually be defined multiple ways. And, incidentally, proving different definitions equivalent is a rather important thing -- for example, it's good to know that we will get correct results when we try to do Euclidean geometry by looking at ordered pairs of real numbers.

You have artificial rigor. How can a mathematician preach about precision yet he is usually late for his lessons. That is just being a hippocrit.
Now you're just being silly! :tongue:

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#### haki

Hurkyl said:
Even from a purely physical perspective... of course there's infinite precision. All of our physical theories are based on there being an "infinite precision" universe out there. E.G. classicaly, a particle has an "infinitely precise" position, whether or not we're capable of measuring it. Quantum mechanically, a state is "infinitely precise", even if we can't hope to know a fraction of the information contained in it.
Ever heard of the uncertainty principle? QM is not a deterministic study but rather an indeterministic study of possibilites.

#### matt grime

Homework Helper
haki said:
I find that contradicting,
why? my response were written about different things. sqrt(2) does not equal 1.41, but that has nothing to do with the other example you gave. Note, I did qualify my statement by saying 'if you had related things correctly'; evidently you hadn't.

you agree that you cannot have higher precision put into the equation as the precision that comes out of the equation.

2 * sqrt(3) is for me of finite precision, 2 is of 1 significat digit precision, and 3 is aswell of 1 SDP

you are confusing maths with physics.

See I even see numbers diferently, A mathematician would see 1 as 1.0000... or an alternative version 0.999999.... but I see it as a 1 SDP number not as a infinite precision number.
again, you are mistaking maths with physics.

Here you are the one who is not writing things the correct way, if 1 is for you 1.0000... then you should have noted that and wrote 1.0 and put a bar over the zero denoting that there are infinity of zeros after the 1 or made another notation

again you are confusing maths with... well never mind, evidently you do not care about the differences...

#### matt grime

Homework Helper
haki said:
What I mean by understaning is this. You should know the meaning of what is it that you are trying to achieve. Take for example this:

32 = 2^x

find the value of X, if you are a Math teacher, just for fun, ask somebody to give you the result. If somebody would understand what it is that you are trying to achive, that is you need to find the value of x, one would answer 5 because (2*2*2*2*2 = 16 * 2 = 32).
You also seem to misunderstand maths. Again. An answer is log32/log2 for absolutely any base of log we choose. You do understand why don't you? Since understanding is 'soo' important. I understand why. I understand why very well. Do you?

But you as a mathematician would probably say that is just GUESSING??? You got the solution using arithmetic and not algebra!
That makes no sense whatsoever.

But hey just doing the logarithms needs no understanding from you. Doing it the way I have show had to do with understanding the problem.
What way have you shown us, precisely?

See if he would understood what was the thing he tried to achive he would have know that you can write
You could attempt to write in grammatically correct sentences. I have no idea what it is you just attempted to say in that 'sentence' .

And the focus is on formalites oh, better say the Mathematical rigor, everything must be rigorus. Why?
So you know it is correct, rather than suspecting it might be correct, perhaps, under someconditions that we don't understand.

Mathematics are wild west. Take a look at geometry. If I recall there is no one "correct"
axiomatization of geometry!! It varies from textbook to textbook!
What? That just means to me that you have no idea of the difference between Euclidean, Hyperbolic, and Spherical geometry. There is no such thing as 'absolutely' correct.

Euclidean geometry is what you want to use if you're drawing up blue prints for a house, spherical if you want to fly plane across the atlantic, and hyperbolic if you want to allow for relativistic events.

And then you dare questioning people on the rigor, yet you don't even have your facts straight!
With respect you are talking crap, bollocks, out of your arse, or you are just plain ignorant, possibly all 4.

For example, If I were to take a physics lessons in Japan or in South Africa the content would be the same
Eh? What are you talking about. The difference is not cultural. If you took a course on quantum physics or solid state physics the courses would be different because they are explaining different things, just as the different types of geometry are explaining different things.

the explanation is bound to differ from professor to professor.
No: Euclidean geometry is the same irrespective of the lecturer.

But the 2nd newton law will be the same everywhere.
and the wave particle duality of light is not the same as the fact that protons have up up down quarks in their make up. So?

Yet if I were to take geometry lessons there are bound to be some differences in the content.
Need I point out that solid state versus quantum mechanics is not the same, again?

If in some scheme you can take something as an axiom and in another sheme you can take that as a theorem, where is the rigor If Math is "universal" as in universally the same, how can there be no correct axiomatization of geometry?

you are the only person saying there is such a thing as 'correct' geometry. To everyone else there are 3 different kinds of geometry depending upon how one chooses to take the parallel postulate, and more importantly there are 'real life' models of all 3.

You have artificial rigor. How can a mathematician preach about precision yet he is usually late for his lessons. That is just being a hippocrit.
If your'e going to hurl insults at least learn to spell them correctly. What has rigour to do with punctuality, anyway?

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#### matt grime

Homework Helper
haki said:
Ever heard of the uncertainty principle? QM is not a deterministic study but rather an indeterministic study of possibilites.
Can you philosophically defend that QM is not deterministic? The particle is where the particle is. We just can't measure it, which is not the same thing at all.

#### Hurkyl

Staff Emeritus
Gold Member
Ever heard of the uncertainty principle?
Sure. It says:

The (infinitely precise) standard deviation of an observable, multiplied by the (infinitely precise) standard deviation of another observable must be at least half the (infinitely precise) expected value of their commutator.

(And, of course, the factor of "one half" is also infinitely precise)

QM is not a deterministic study but rather an indeterministic study of possibilites.
If you think it's that obvious, then you don't understand QM.

#### haki

matt grime said:
Can you philosophically defend that QM is not deterministic? The particle is where the particle is. We just can't measure it, which is not the same thing at all.
Can you prove that the particle has a definite position!? You assume that a particle ought to have a determined position since that is what our human experience would have believed us to be. Aha! If QM would be deterministic then you would be able to explain us the infamous double slit experiment. Please explain it to us. If a particle has a definite position why does the interference pattern emerge and not what our human experience would have assumed? You seam to have all the answers try to answer this one.

#### Hurkyl

Staff Emeritus
Gold Member
haki said:
Can you prove that the particle has a definite position!? You assume that a particle ought to have a determined position since that is what our human experience would have believed us to be. Aha! If QM would be deterministic then you would be able to explain us the infamous double slit experiment. Please explain it to us. If a particle has a definite position why does the interference pattern emerge and not what our human experience would have assumed? You seam to have all the answers try to answer this one.
The Bohm interpretation is a counterexample to your position.

I hope nobody minds, but I'm going to lock this thread -- it seems that the original discussion has been exhausted, and now people are just trying to "score points". If you really want to discuss philosophy of QM, feel free to start a thread in the QM section, or in the philosophy of science section. (The former is better, methinks)

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#### HallsofIvy

Homework Helper
Then clearly, what you should do is go to your teachers, chair of the department, and perhaps the president of the college, and explain clearly exactly what they are to teach you and how! Since they will immediately, if they haven't already, recognize that you already know far more than they do, they certainly would lose no time in complying.

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