accepted mathematical problem
I found some problems that i doubly call them a mathematics competition problems, one example is:
Problem 1:
"The autopsion result that propofol was found in the death body of the king of pop, Michael Jackson.
The problem was given as a take-home problem for a math competition. From my opinion, this is not a typical of the competition problem nor a research one (it's just a new story taken from CNN.)
It has no objection, no clear instruction, even worse: given that we know what to do, it does not provide us with a sufficient data to proceed (even though we can search it ourselves.)
Problem 1 might be transformed into a mathematics problem. But from the way it was stated, i think it is not a good mathematics problem, not even i want to call it one.
Without dishonor the numerical analysts and staticians, I prefer the problem with the exact answer. That is whenever the problem is
being attacked by some mathematicians, then they would purpose a same true answer (up to any dummy variables or the notations.)
If the numerical analysts and staticians are being included, then our type of problem must tackle the ambiguity of its purposes and it should possess a halt-condition (the condition that must be satisfied to flag the problem as SOLVED.)
[The Problem 1 above failed to win these criteria]
One might thinks that a yes-no (true-false) question would fit this criteria, but others type of problem like root-finding problem also fit the criteria well.
Thinking about this type of problems, lets make a deeper investigation about how they might look like.
How about this problem:
Problem 2 :\[ \text{"Find $x$!"}\]
First question might be arise here would be "What is the x?". Although "Find x!" is a question, but it shouldn't be called a math question, since it is not a proposition. That's right, we need a proposition (you can find $x$ by pointing out the variable ;p). The question itself must be related to a proposition.
Now how about:
Problem 3 :\[\text{"The production of the gasoline has arised up to 110.3$\%$. True or Not?"}\]
Here we have the question and the proposition. But what's wrong? well it might be a well-posed problem, but it can only be answered, given a sufficient and relevant data to proceed.
Another lesson here: The problem which doesn't give a complete data can lead to a dead-end.
Perhaps, some theorists might try to independently collect the data in order to solve the problem, or simply by making some assumptions before they proceed. But then the answers can be varied as these assumptions are being made.
As we already noted before, we want a same correct answer for each people who solve the problem. So we need to include the general assumption into the problem. The stated assumption simply becomes a signed proposition, that is the proposition which value is known.
The proposition which has an unknown value is included in the question. Notice that as long as we know its logic value (or simply assign it by using the assumption), the proposition itself can be viewed as a data. So we count them as one.
Are those all we need?
Oh no, what about the Definitions and Axioms? No worry, they are also can be viewed as propositions. So, let test our result:
Problem 4:\[\text{ Given integer $n \geq 3$. Find all positive integers $x,y$ and $z$ such that} \] \[x^n+y^n=z^n\]
The signed propositions here are :
1. $n$ is an integer greater than 3
2. $x,y$ and $z$ are positive integers.
The above propositions need another propositions (i.e definition of integers or axiom of numbers.)
And the question is "Find all $x,y,z$ such that the proposition $x^n+y^n=z^n$ is true".
The proposition and the question, that's all we need?
Back to 1920s, the structure about how (in general) the mathematics is posed was already stated by a famous German Mathematicians , David Hilbert. It's called the Hilbert program. The goal is to formalize the mathematics such that :
Well, not all of them have.
The first Godel's Incompleteness Theorem merely state that
"There exists an arithmatical statement that is true but not provable."
Some mathematicians argue this "scary" statement might be exists in the unintended area of mathematics.
For example, in the number theory, since the area relies on the arithmatic statements. There could be some number-theorytic problem that is true but cannot be proved.
Does the Godel Sentence only exists on the alien area in mathematics?
Not really, The Whitehead Problem which states
Is every abelian group $G$ with $Ext(G,\mathbb{Z})=0$ a free abelian group?
Where $\mathbb{Z}$ is the group of integers and $Ext$ is the Ext Functor. This Whitehead problem was proved by shelah to be undecideable within the Zermelo–Fraenkel set theory. That is it cannot be proved to be true or wrong given the already known axiom in Zermelo–Fraenkel set theory.
Problem 1:
"The autopsion result that propofol was found in the death body of the king of pop, Michael Jackson.
Dr Murray told the police that he gave the propofol in effort to help him sleep. Worried that jackson might become addicted to
the drug, Murray tried to wean Jackson from it and using another alternative drugs to make Jackson sleep.
But On the morning, Jackson had found to be death."
Formulate and solve the mathematics model of the above problem!
the drug, Murray tried to wean Jackson from it and using another alternative drugs to make Jackson sleep.
But On the morning, Jackson had found to be death."
Formulate and solve the mathematics model of the above problem!
The problem was given as a take-home problem for a math competition. From my opinion, this is not a typical of the competition problem nor a research one (it's just a new story taken from CNN.)
It has no objection, no clear instruction, even worse: given that we know what to do, it does not provide us with a sufficient data to proceed (even though we can search it ourselves.)
Problem 1 might be transformed into a mathematics problem. But from the way it was stated, i think it is not a good mathematics problem, not even i want to call it one.
Without dishonor the numerical analysts and staticians, I prefer the problem with the exact answer. That is whenever the problem is
being attacked by some mathematicians, then they would purpose a same true answer (up to any dummy variables or the notations.)
If the numerical analysts and staticians are being included, then our type of problem must tackle the ambiguity of its purposes and it should possess a halt-condition (the condition that must be satisfied to flag the problem as SOLVED.)
[The Problem 1 above failed to win these criteria]
One might thinks that a yes-no (true-false) question would fit this criteria, but others type of problem like root-finding problem also fit the criteria well.
Thinking about this type of problems, lets make a deeper investigation about how they might look like.
How about this problem:
Problem 2 :\[ \text{"Find $x$!"}\]
First question might be arise here would be "What is the x?". Although "Find x!" is a question, but it shouldn't be called a math question, since it is not a proposition. That's right, we need a proposition (you can find $x$ by pointing out the variable ;p). The question itself must be related to a proposition.
Now how about:
Problem 3 :\[\text{"The production of the gasoline has arised up to 110.3$\%$. True or Not?"}\]
Here we have the question and the proposition. But what's wrong? well it might be a well-posed problem, but it can only be answered, given a sufficient and relevant data to proceed.
Another lesson here: The problem which doesn't give a complete data can lead to a dead-end.
Perhaps, some theorists might try to independently collect the data in order to solve the problem, or simply by making some assumptions before they proceed. But then the answers can be varied as these assumptions are being made.
As we already noted before, we want a same correct answer for each people who solve the problem. So we need to include the general assumption into the problem. The stated assumption simply becomes a signed proposition, that is the proposition which value is known.
The proposition which has an unknown value is included in the question. Notice that as long as we know its logic value (or simply assign it by using the assumption), the proposition itself can be viewed as a data. So we count them as one.
Are those all we need?
Oh no, what about the Definitions and Axioms? No worry, they are also can be viewed as propositions. So, let test our result:
Problem 4:\[\text{ Given integer $n \geq 3$. Find all positive integers $x,y$ and $z$ such that} \] \[x^n+y^n=z^n\]
The signed propositions here are :
1. $n$ is an integer greater than 3
2. $x,y$ and $z$ are positive integers.
The above propositions need another propositions (i.e definition of integers or axiom of numbers.)
And the question is "Find all $x,y,z$ such that the proposition $x^n+y^n=z^n$ is true".
The proposition and the question, that's all we need?
Back to 1920s, the structure about how (in general) the mathematics is posed was already stated by a famous German Mathematicians , David Hilbert. It's called the Hilbert program. The goal is to formalize the mathematics such that :
- Completeness: a proof that all true mathematical statements can be proved in the formalism
- Consistency: a proof that no contradiction can be obtained in the formalism of mathematics.
- Conservation: a proof that any result about "real objects" obtained using reasoning about "ideal objects" (such as uncountable sets) can be proved without using ideal objects.
- Decidability: there should be an algorithm for deciding the truth or falsity of any mathematical statement.
Well, not all of them have.
The first Godel's Incompleteness Theorem merely state that
"There exists an arithmatical statement that is true but not provable."
Some mathematicians argue this "scary" statement might be exists in the unintended area of mathematics.
For example, in the number theory, since the area relies on the arithmatic statements. There could be some number-theorytic problem that is true but cannot be proved.
Does the Godel Sentence only exists on the alien area in mathematics?
Not really, The Whitehead Problem which states
Is every abelian group $G$ with $Ext(G,\mathbb{Z})=0$ a free abelian group?
Where $\mathbb{Z}$ is the group of integers and $Ext$ is the Ext Functor. This Whitehead problem was proved by shelah to be undecideable within the Zermelo–Fraenkel set theory. That is it cannot be proved to be true or wrong given the already known axiom in Zermelo–Fraenkel set theory.
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