logical mathematics theory

[Juilingstar177]

x = x can be termed as one of the most basic and fundamental law ofarithematical mathematics. This law can easily be proved as.

Let x be not equal to x.

then let x=1 and x=2

but 1 is not equal to 2 (logics)

hence, x=x.

 

Don't you feel the power of this equation? Even of it's simplitiy, if thisproved wrong, will thorougly devastate the present treatment of mathematic,making each and every constant power to attain a variablic value.

 

x=x and x+ function=y : x = y or x is not equal to y

 

made the whole arithematic mathematics. We build on these two basic axiomsand made what see as a mathematics. Logical Deduction.

 

Logical Deduction were building blocks of Arithematics. We made different typesof functions, we found different values of x or y and we are still building on.But the basic problem in the modern mathematics we suffer is lack of logics.Logics are only limited to axioms and postulates, and logical deduction is nowtreated different from mathematics. This undifferent treatement have greatedbig holes in our treatment of mathematics.

 

One of the biggest problem is treatement of value of pi. Logically, it is notpossible to find the value of pi. This is because of the following logicaldeduction that even if we go to atomic level we will find that circles are notcircles. They are just a polygon with finite no. of side. Hence the value of piis not defined and can never be found. But this logical deduction could not betreated in mathematical concepts.

 

Also, if we divide 10 apples among 0 people, we get 10 apples. Won't we?Obviously. Let's take a simple example. Consider you give 10 people a party.But unfortunately, no one comes. So, the apples you kept for them remains thesame won't they? But Mathematics treat it undefined.

 

So, we are moving away from Logics in Mathematics. We are just pushing awayour fondations by not agreeing logical treatments in mathematics, and givingthem more priority over other mathematical concepts.

 

 

 

 

 

 

 

 

 

 

 

[mathematic]

"One of the biggest problem is treatement of value of pi. Logically, it is notpossible to find the value of pi. This is because of the following logicaldeduction that even if we go to atomic level we will find that circles are notcircles. They are just a polygon with finite no. of side. Hence the value of piis not defined and can never be found. But this logical deduction could not betreated in mathematical concepts."

 

You are mixing mathematics with physics. Pi is a well defined mathematical concept, based on mathematical description of circles. Any drawing at any level will be an approximation.

[DrRocket]

"One of the biggest problem is treatement of value of pi. Logically, it is notpossible to find the value of pi. This is because of the following logicaldeduction that even if we go to atomic level we will find that circles are notcircles. They are just a polygon with finite no. of side. Hence the value of piis not defined and can never be found. But this logical deduction could not betreated in mathematical concepts."

 

You are mixing mathematics with physics. Pi is a well defined mathematical concept, based on mathematical description of circles. Any drawing at any level will be an approximation.

 

Pi can be, and is, defined in modern mathematics independently of any geometric construction, though pi is the ratio of the circumference of the circle to its diameter.

 

Herewith is the modern analytic definition:

 

The complex exponential function defined by the power series

 

[math] e^{z} = \displaystyle \sum_{n=0}^\infty \frac {z^n}{n!}[/math]

 

can be shown to be periodic (see e.g. Rudin's Real and Complex Analysis for an elegant proof). [math]\pi[/math] is defined to be the period of this function divided by 2i.

 

This gives you a definition traceable to the Zermelo Fraenkel axioms and quite independent of Euclidean geometry.

 

If this is not familiar, recall the Euler formula [math]e^{i \theta}= cos \theta \ + \ i \ sin \theta [/math] .

[mathematic]

Pi can be, and is, defined in modern mathematics independently of any geometric construction, though pi is the ratio of the circumference of the circle to its diameter.

 

Herewith is the modern analytic definition:

 

The complex exponential function defined by the power series

 

[math] e^{z} = \displaystyle \sum_{n=0}^\infty \frac {z^n}{n!}[/math]

 

can be shown to be periodic (see e.g. Rudin's Real and Complex Analysis for an elegant proof). [math]\pi[/math] is defined to be the period of this function divided by 2i.

 

This gives you a definition traceable to the Zermelo Fraenkel axioms and quite independent of Euclidean geometry.

 

If this is not familiar, recall the Euler formula [math]e^{i \theta}= cos \theta \ + \ i \ sin \theta [/math] .

 

Pi can be defined either way, depending on your taste. ZF or Euclid.

[the tree]

Pi can be defined either way, depending on your taste. ZF or Euclid.

Although it is very, very, very important to know that the two definitions are equivalent.

 

@Juilingstar177, I would strongly recommend that if you are interested in logic then you begin to study it properly. It is a rich, beautiful, complex and powerful field of study. But you will not be able to study it just by making assumptions and running with them, actually you wont be able to study anything that way.

 

For starters, you trip up fairly early on, trying to prove a definition (which is somewhat like trying to scribble over a bucket of black paint with a black pen in order to make it black).

x = x can be termed as one of the most basic and fundamental law of [arithmetic]. This law can easily be proved as...

Actually, no, there's nothing to be proved - it's just part of the definition of "=". It's called reflexivity and alongside symmetry and transitivity we can define an equivalence relation which "=" is the canonical example of and yes a pretty damn important tool in mathematics but a pretty uninteresting one on the whole.

 

One of the biggest problem is treatement of value of pi. Logically, it is notpossible to find the value of pi. This is because of the following logicaldeduction that even if we go to atomic level we will find that circles are notcircles. They are just a polygon with finite no. of side. Hence the value of piis not defined and can never be found. But this logical deduction could not betreated in mathematical concepts.

I think this has been fairly well covered. Just don't mix up physics and mathematics - in short mathematics is logical and the universe is well, not.

 

Also, if we divide 10 apples among 0 people, we get 10 apples. Won't we?Obviously. Let's take a simple example. Consider you give 10 people a party.But unfortunately, no one comes. So, the apples you kept for them remains thesame won't they? But Mathematics treat it undefined.

I would strongly advise you to simply research the definition of division in the context of mathematics, you'll be kicking yourself fairly quickly.

 

And try not to make analogies about apples - only one guy ever gained any credibility doing that and he's generally considered a bit a dick nowadays anyway.

 

So, we are moving away from Logics in Mathematics.

Actually far from it, formal logic has had a much more significant role in mathematics in the past century than it has ever had since the days of Plato, I'm really surprised people aren't more aware of this - just think what all the software on your computer is made out of!

 

If you really want start appreciating the places that logic and mathematics can take you - then start asking questions, reading, thinking, asking more questions and then challenging the answers you receive.

 

You can of course declare simply how you think things are and feel smart because you used the word "logic" (which, for the record, a lot of us hear every day). But if you take a moment to learn how things actually are then prepare to have your mind blown, because they are so much bigger and so much more complex than any of us. You will never, ever want to go back to living in the dark.

[khaled]

I agree with the tree .. All branches of science depends on mathematics, and mathematics depends on Logic

 

.. in the past century, the study of Logic, Proof, & Formal Systems emerged, and so at one time men as Kurt Godel and David Hilbert,

 

gathered unto the idea that mathematics shouldn't be scattered all over, that mathematics should be a unified formal system where

 

there exist no contradictions (P ^ not P) .. thus to add anything new to mathematics, you need to prove it, and so Proof is

 

an important aspect now.

 

Mathematics and Logic are important to me, because I'm a computer scientist (mathematical logic scientist)

Edited January 29, 2012 by khaled

[the tree]

I agree with the tree .. All branches of science are based on mathematics, and mathematics is based on Logic

Okay, I'm just going to be tearing my hair out now.

[khaled]

Okay, I'm just going to be tearing my hair out now.

 

Did I say something wrong, maybe I should've used the word "depends" rather than "based" .. I've just modified my post

 

Still I can't understand why would a mathematician tear off his hair, on what I've said.

[1=1]

Did I say something wrong, maybe I should've used the word "depends" rather than "based" .. I've just modified my post

 

Still I can't understand why would a mathematician tear off his hair, on what I've said.

 

I think science and mathematics are based on logic. Some sciences aren't based entirely off maths, and use logic to establish an understanding on the topic.

[DrRocket]

Did I say something wrong, maybe I should've used the word "depends" rather than "based" .. I've just modified my post

 

Still I can't understand why would a mathematician tear off his hair, on what I've said.

 

You haven't said anything right.

 

I too am surprised that a mathematician would tear off his hair. However, I do understand why he might chew off his arm.

[khaled]

You haven't said anything right.

 

If you math experts tell me that things I'm told by my math professor are wrong, and you are making jokes,

 

rather than clearing up things and saying "what is wrong" .. how can I know what is right then ?!

[DrRocket]

If you math experts tell me that things I'm told by my math professor are wrong, and you are making jokes,

 

rather than clearing up things and saying "what is wrong" .. how can I know what is right then ?!

 

The key to understanding mathematics is being able to read and understand it and do the proofs for yourself.

 

That way, by applying rigor and logic, you know what is right without having to rely on anyone, professors included, to tell you what is right. In short you figure out what is right for yourself -- based on what you can prove, not what you "think" or "believe".

[khaled]

I've once read "Observe .. Think .. Propose a theory .. prove your theory"

 

.. and "there exist facts that we know they're true, but we can't prove them"

 

One theory says "anything that can happen, will happen" which was proved under the scope of nature ..

 

but if we interpret it mathematically, we get that "there exist no event with 0 % probability" which

 

I really don't know how to start thinking how to disprove .. I remember taking the class on Probability,

 

Statistics, and Modeling .. where normalized probability space is where we have an assumption that

 

initially "everything happened once".

Edited January 30, 2012 by khaled

[DrRocket]

I've once read "Observe .. Think .. Propose a theory .. prove your theory"

 

.. and "there exist facts that we know they're true, but we can't prove them"

 

One theory says "anything that can happen, will happen" which was proved under the scope of nature ..

 

but if we interpret it mathematically, we get that "there exist no event with 0 % probability" which

 

I really don't know how to start thinking how to disprove .. I remember taking the class on Probability,

 

Statistics, and Modeling .. where normalized probability space is where we have an assumption that

 

initially "everything happened once".

 

Everything in this paragraph is either a) wrong or b) nonsensical.

 

"Anything that can happen, will happen" is a very rough translation of the Law of Large Numbers which implies "that any event of positive probability will occur infinitely often in infinitely many independent trials, with probability one".

 

However it most certainly does not imply that there is no event of 0 probability. It does not even imply that an event of probability 0 cannot occur, but only that the occurrence has probability 0. And no, "probability 0 does NOT mean that an occurrence is impossible."

 

In fact, loosely speaking the probability of an event [math]A[/math] is [math]\displaystyle \lim_{n \to \infty} \frac {number \ of \ occurrences \ of \ A}{n}[/math] where [math]n[/math] is the number of trials. If, for instance, there are only a finite number of occurrences of [math]A[/math] in infinitely many trials then the probability of[math] A[/math] is zero, even though it did occur. Note also that in order to have a non-zero probability that [math]A [/math] must occur infinitely many times in infinitely many trials, as indicated above. This is made much more precise in a proper treatment of probability theory based, following Kolmogorov, on measure theory.

 

The statement that "there exist facts that we know they're true, but we can't prove them" is a very bad paraphrase of Godel's first incompleteness theorem, which in fact states that, given any system of axioms that admits ordinary arithmetic there will be true theorems that are not provable by first-order logic. That does not necessarily mean that "we know that they're true" since knowledge in mathematics comes from proof and if we know it is true that means that we can prove it. It also does not mean that statements not provable via first-order logic are not provable via higher-order methods. Moreover, there are assertions, such as the Axiom of Choice and the Continuum Hypothesis that are known to be independent of the basic Zermelo Fraenkel axioms.

 

There is no such thing as a "normalized probability space", though there is such a thing as the "normal probability density" also called the "Gaussian probability density" which figures rather prominently in the theory -- Google Central Limit Theorem. The notion that "everything happens at once" makes no sense in this context. Theere is in fact no need to normalize a probabilty space since, by definition, the total measure of a probability space is always 1.