## Is all math essentially: "Solve for x?"

 Quote by I like Serena But... a,b, c are integers... didn't I mention that they are elements of the (apparent) same ##\Bbb N## that you are referring to?

Nop. You wrote "The problem can be rephrased as: find integers a, b, c, and n > 2 such that $\,a^n+b^n=c^n\,.$"

And then I wrote what I wrote. :)

DonAntonio

 Quote by Borek Proof that for n > 2 there are no integers a, b and c that satisfy the equation an+bn=cn is hardly of the kind "solve for x".
For sake of discussion I'll take the opposite side of that.

Find integers n > 2, a, b, c such that a^n + b^n = c^n or else show that no such integers can exist

That's a math problem in which we're asked to find four unknowns that satisfy a condition. In that respect it's no different than being asked to find x given that 2x = 5 (or show that no such x exists. Maybe we're in the integers.)

We state a condition or relationship among some variables; then we're asked to find particular numbers that satisfy that relationship, or prove that none exist.

You could do the same thing for the Riemann hypothesis. Find x such that x is a zero off the critical line, etc.

 no. "solve for X" is science. Math is much more.

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 Quote by chill_factor no. "solve for X" is science. Math is much more.
A scientist might claim it's the other way around. ;)

 Quote by I like Serena But... a,b, c are integers... didn't I mention that they are elements of the (apparent) same ##\Bbb N## that you are referring to?
0 is an integer. The set of integers is ##\Bbb Z##.

 Blog Entries: 1 Although many math problems will ask you to solve one or more variables, while in class, it is about understanding WHAT that variable means. What each variable you used on the way means. Math in the higher levels is more about understanding what it represents (as well as solving for x). - For example, you'll run into classes like Calculus with Analytical Geometry where you learn what the geometric representation/meaning is of the different Calculus operations. In applied mathematics...yes, solve for x :D Sorry if this is a tad bit off =P I'm only a freshman in college.
 No, "solving for x" as you know it forms just a little bit of math. Your criticism of the Riemann Hypothesis is also wrong. We can find zeroes of the Riemann zeta function relatively easily, especially with the computers of our age. The problem is that where those zeroes actually lie. After all, we cannot compute infinitely many zeros with a computer, so one needs a proof to conclude all zeros lie on the critical line. Even if you do think this is solving for X (which you shouldn't), the methods of solving are still much different. You will change your mind if you reach the level in which you can understand the Riemann Hypothesis and what it actually means.

 Quote by Millennial No, "solving for x" as you know it forms just a little bit of math. Your criticism of the Riemann Hypothesis is also wrong. We can find zeroes of the Riemann zeta function relatively easily, especially with the computers of our age. The problem is that where those zeroes actually lie. After all, we cannot compute infinitely many zeros with a computer, so one needs a proof to conclude all zeros lie on the critical line. Even if you do think this is solving for X (which you shouldn't), the methods of solving are still much different. You will change your mind if you reach the level in which you can understand the Riemann Hypothesis and what it actually means.
Was that for me? I wasn't able to tell if you're responding to what I wrote. If so, I'll be happy to explain. RH is the classic "find x" problem. Of course it's at a much higher level of mathematical sophistication than finding the solution of 2x = 5 in the integers (or proving no solution exists); but it's precisely the same in terms of structure.

Let's examine the structure of "Find an integer x such that 2x = 5, or prove no such exists."

We have a predicate P. In this case, P (x) is "x is an integer and 2x = 5."

We are asked to find any and all x that satisfy P, or else prove that no such x can exist.

An exhaustive computer search proves fruitless. 2*1 = 2, 2*2 = 4, 2*3 = 6, 2*(-180) = -360, ... hmmm, this won't work. I need some theory.

I consult a local mathematician. (A local mathematician is someone who appears to be a mathematician within some neighborhood, but who may or may not be a mathematician when viewed from a global perspective. A lot like many of the denizens of online math forums!) She walks me through some elementary number theory and some elementary ring theory, then points out that on the left, if x is an integer, then 2x must be an element of the ideal generated by 2, denoted <2>. But 5 is not an element of this ideal, because 5 = 2(2) + 1 and 1 is not an element of <2>.

I am enlightened. We have used theory to show that no such integer x exists satisfying 2x = 5.

Now I start wondering about RH. Again we have a predicate P(x), which in this case is "x is a nontrivial zero of the zeta function and the real part of x is not 1/2".

Of course P is a fairly complicated predicate, and it would take some time to boil it down to a form that would satisfy a logician's definition of a predicate. But it can be done. We have a theory that says so :-)

Now we are challenged to find x in the complex numbers that satisfies P, or else show that no such x can exist.

In other words, RH is far more mathematically sophisticated than 2x = 5. For one thing, nobody's offering a million dollars for a solution of the famous 2x = 5 problem. But the form is identical. RH is a classic "find x" problem. You have a predicate and you're asked to find the value(s) that satisfy P, or else show that no such x can exist.

In fact, many math problems have this structure. Take the famous classification of finite, simple groups. In this case, x is a classification. A classification in this context is a collection of groups such that every finite simple group is isomorphic to exactly one member of the collection. If we were sufficiently motivated, we could write down a predicate that encapsulates the meaning of "classification." Again, we have a classical "find x" problem.

It's surprising just how many deep mathematical problems have the same structure: "Find all x satisfying predicate P, or prove that no such exists."

If you are building a theory, finding x does not typically come into play. But whenever you are solving a problem, "finding x" is typically the exact form of the problem. We're looking for elements of some set that satisfy a predicate.

And as I pointed out, exhaustive computer search is no more effective in solving 2x = 5 than it is in solving RH; so the fact that a given problem requires theory is no objection at all.

 Quote by SteveL27 Was that for me? I wasn't able to tell if you're responding to what I wrote. If so, I'll be happy to explain. RH is the classic "find x" problem. Of course it's at a much higher level of mathematical sophistication than finding the solution of 2x = 5 in the integers (or proving no solution exists); but it's precisely the same in terms of structure.

I don't this is even close to be true. You're oversimplifying things in such a way that almost all on this universe, mathematics

or not, could be put in a "find x" problem:

1) What is the recession problem? Find x= money problem

2) What is world's overpopulation problem? Find x=good birth control problem

3) What is the world's wars problem? Find x= a good reason for peoples to love each other

4) What is the international terror problem? Find x=the terror cells and destroy them, or look (3) above

5) What' the P = NP problem? Find x=a problem where P doesn't equal NP...

The above sounds really absurd and even ridiculous, yet that's what we'd get to if we'd accept your approach.

 Let's examine the structure of "Find an integer x such that 2x = 5, or prove no such exists." We have a predicate P. In this case, P (x) is "x is an integer and 2x = 5." We are asked to find any and all x that satisfy P, or else prove that no such x can exist. An exhaustive computer search proves fruitless. 2*1 = 2, 2*2 = 4, 2*3 = 6, 2*(-180) = -360, ... hmmm, this won't work. I need some theory. I consult a local mathematician. (A local mathematician is someone who appears to be a mathematician within some neighborhood, but who may or may not be a mathematician when viewed from a global perspective. A lot like many of the denizens of online math forums!) She walks me through some elementary number theory and some elementary ring theory, then points out that on the left, if x is an integer, then 2x must be an element of the ideal generated by 2, denoted <2>. But 5 is not an element of this ideal, because 5 = 2(2) + 1 and 1 is not an element of <2>. I am enlightened. We have used theory to show that no such integer x exists satisfying 2x = 5. Now I start wondering about RH. Again we have a predicate P(x), which in this case is "x is a nontrivial zero of the zeta function and the real part of x is not 1/2".
Perhaps this predicate would fit the problem "Disprove RH" instead of "prove RH": how would you put RH in a "find x" format?

 Of course P is a fairly complicated predicate, and it would take some time to boil it down to a form that would satisfy a logician's definition of a predicate. But it can be done. We have a theory that says so :-) Now we are challenged to find x in the complex numbers that satisfies P, or else show that no such x can exist. In other words, RH is far more mathematically sophisticated than 2x = 5. For one thing, nobody's offering a million dollars for a solution of the famous 2x = 5 problem. But the form is identical. RH is a classic "find x" problem.
No, it's not. In the best of the cases, disproving it would be.

 You have a predicate and you're asked to find the value(s) that satisfy P, or else show that no such x can exist. In fact, many math problems have this structure. Take the famous classification of finite, simple groups. In this case, x is a classification. A classification in this context is a collection of groups such that every finite simple group is isomorphic to exactly one member of the collection. If we were sufficiently motivated, we could write down a predicate that encapsulates the meaning of "classification." Again, we have a classical "find x" problem.
Are you being serious or you're just pulling our legs? So we could say as well that someone having a huge problem because his son

is ill is just having a "find x" problem, this time x being "a cure, a drug, an operation"...? Common.

The problem of classification of finite groups was FIRST reduced to classify all finite simple groups. To call this problem a "find x"

problem with x = classification, absolutely trivializes ANY human endevour in ANY field.

 It's surprising just how many deep mathematical problems have the same structure: "Find all x satisfying predicate P, or prove that no such exists."
I think it is more surprising you actually believe this...but there's lots of weird/funny/amazing beliefs, so...

DonAntonio

 If you are building a theory, finding x does not typically come into play. But whenever you are solving a problem, "finding x" is typically the exact form of the problem. We're looking for elements of some set that satisfy a predicate. And as I pointed out, exhaustive computer search is no more effective in solving 2x = 5 than it is in solving RH; so the fact that a given problem requires theory is no objection at all.

 DonAntonio: One comment about your above situations. In a lot of these situations not only is the problem not specified adequately, but the known information, assumptions, and constraints are not even close to being specified adequately either. You could formulate these problems mathematically, but it would mean at a minimum that you take the contextual knowledge that we take for granted and quantify and represent it accurately. This means finding a way to "un-project" everything and then evaluate it in a way that allows a kind of deduction and inference that we do so easily, but that is hard to do using our current developed mathematical frameworks. The other thing is that in mathematics, most people can't deal with more than say around 7 or eight variables at a time in working memory and when analysis takes place (I'm talking about the general activity of breaking things down, not just a mathematical decomposition) a lot of information is "projected away" to make things simpler. The other thing that's even more critical is the nature between cause and effect of a problem. Mathematically a lot of the problems we deal with have very explicitly formulations that are quantifiable (i.e can be calculated) where-as the problems you have described have no real solid definition in terms of how the variables are related (even in a stochastic sense). So ultimately you can't compare the situations that mathematicians, scientists, and engineers face with those kinds of problems. The un-projected system however (even if that system was somewhat a complete description in all respects, and I know this is a very generous assumption) is going to be infeasible when it comes to using our tools vs the natural tools that have been afforded to us by nature herself.

 Quote by DonAntonio I think it is more surprising you actually believe this...but there's lots of weird/funny/amazing beliefs, so...
If this is a question of faith for you, then we could stop the discussion right here and now. If not, would you be able to show at least one theorem or a conjecture that cannot be cast in the predicate form I mentioned in #13 in this thread?

 Quote by chiro DonAntonio: One comment about your above situations. In a lot of these situations not only is the problem not specified adequately, but the known information, assumptions, and constraints are not even close to being specified adequately either.


Of course: that's part of my point. Many of those situations are specified in such a way as stating that RH is "find a x solution to the

zeta functions with real part not equal to 12"...do you really think this expresses the problem in a sound, nice way?
 You could formulate these problems mathematically, but it would mean at a minimum that you take the contextual knowledge that we take for granted and quantify and represent it accurately.


I don't want to pose them mathematically: that was another part of my point, namely to show that stretching the argument we

could pose practically ANY problem on the same lame "find x" line.
 This means finding a way to "un-project" everything and then evaluate it in a way that allows a kind of deduction and inference that we do so easily, but that is hard to do using our current developed mathematical frameworks. The other thing is that in mathematics, most people can't deal with more than say around 7 or eight variables at a time in working memory and when analysis takes place (I'm talking about the general activity of breaking things down, not just a mathematical decomposition) a lot of information is "projected away" to make things simpler. The other thing that's even more critical is the nature between cause and effect of a problem. Mathematically a lot of the problems we deal with have very explicitly formulations that are quantifiable (i.e can be calculated) where-as the problems you have described have no real solid definition in terms of how the variables are related (even in a stochastic sense). So ultimately you can't compare the situations that mathematicians, scientists, and engineers face with those kinds of problems.


I think you completely missed my message's point, which was to criticize the stance that "mathematics reduces to "find x"

problems", although later it was changed a little to "solving problems in mathematics reduced to "find x" situations".

I never intended to compare situations, of course. I tried to show that stretching that argument we'd get ridiculous and/or absurd

situations in practically any realm of life, not only mathematics.

DonAntonio
 The un-projected system however (even if that system was somewhat a complete description in all respects, and I know this is a very generous assumption) is going to be infeasible when it comes to using our tools vs the natural tools that have been afforded to us by nature herself.

 Quote by voko If this is a question of faith for you, then we could stop the discussion right here and now. If not, would you be able to show at least one theorem or a conjecture that cannot be cast in the predicate form I mentioned in #13 in this thread?

You, or someone else, already did: RH and the classification of all (simple or not) finite groups cannot be reduced to "find x" in a

sound mathematically way, not can any problem of the kind "show that this group is solvable/nilpotent/abelian/etc."

DonAntonio

 Quote by DonAntonio You, or someone else, already did: RH and the classification of all (simple or not) finite groups cannot be reduced to "find x" in a
This was not the question I asked. I asked for a theorem that cannot be cast in a specific predicate form. The Riemann hypothesis can be cast in that form.

"Solve for x" is merely an interpretation of this form, a valid one at that. If that is an interpretation you dislike for some reason, you should say just this, but you should not say that this interpretation is flawed.

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 Quote by DonAntonio nor can any problem of the kind "show that this group is solvable/nilpotent/abelian/etc."
I'm just picking one:
Show that the group ##\Bbb Z## is abelian.

This can be rephrased as:
Find the set of pairs $X \subseteq \Bbb Z \times \Bbb Z$ that do not commute in the group ##\Bbb Z##.
In particular, if this set X is empty, we call the group abelian.

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 Quote by I like Serena I'm just picking one: Show that the group ##\Bbb Z## is abelian. This can be rephrased as:Find the set of pairs $X \subseteq \Bbb Z \times \Bbb Z$ that do not commute in the group ##\Bbb Z##.In particular, if this set X is empty, we call the group abelian.