Is all math essentially: Solve for x?

[Tyrion101]

Is all math essentially: "Solve for x?"

I am still learning basic math, and in one form or another it all seems basically: "Find x, or in some case some other letter." Or does it become different when you get to the higher maths?

 

[DonAntonio]

I am still learning basic math, and in one form or another it all seems basically: "Find x, or in some case some other letter."

$${}$$
Not even close

Or does it become different when you get to the higher maths?

Very, very different...fortunately enough: much more challenging, interesting, beautiful, astonishing...although, of course, now and then we still have to tackle a "find x such that so and so" question.

DonAntonio

 

[Mentallic]

I am still learning basic math, and in one form or another it all seems basically: "Find x, or in some case some other letter." Or does it become different when you get to the higher maths?

I'm in uni and still finding x's or some other kind of value. Although, the x's I'm typically finding aren't numbers, they're matrices, vectors, functions etc.
But that's just the start of it. There are so many other problems that'll you'll also need to tackle - and likely a lot of times these techniques you've learned will be used in order to help you find x, but also, they might not be.

I'm not sure how basic the basic math you're talking about is, but have you learned about expanding factors yet? Such as a(x+y)=ax+ay? This kind of thing is pretty much as big in maths as finding x is.

 

[voko]

Mathematics is useful because it helps us solve some problems. In such a situation, yes, it is very frequently about "finding x that satisfies these conditions". But this is mostly known as "applied mathematics'. Pure mathematics is about building theories about properties and relations of things that may or may not be useful in applications. Note, however, that there is no clear-cut division between pure and applied. Often, it is a matter of attitude. Plus what is known as pure and even considered not ever likely to become applied, can become applied one day. For example, modern cryptography is almost entirely based on (once) purest branch of mathematics: number theory.

 

[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".

 

[xodin]

I am still learning basic math, and in one form or another it all seems basically: "Find x, or in some case some other letter." Or does it become different when you get to the higher maths?

Similarly to what others have said, higher math often times doesn't even use numbers--they keep it all in variables. For example, in basic calculus you might be given a function of x, and your job is to solve for its derivative or integral, which is still just a function of x. Sure you could plug numbers into anyone of those functions, but that's not always the goal.

 

[phinds]

$${}$$
Not even close

I had to laugh at that 'cause it's EXACTLY what my reaction was when I saw the question. I guess it would be for most folks who've had much exposure to math but it still tickled me to see you took the words right out of my mouth.

 

[voko]

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".

It's a qualified answer that for n > 2 "I cannot solve for x".

 

[phinds]

It's a qualified answer that for n > 2 "I cannot solve for x".

That seems like quite an odd way of looking at it.

 

[voko]

That seems like quite an odd way of looking at it.

As I wrote earlier, applied vs pure is a matter of attitude.

 

[I like Serena]

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".

It's a qualified answer that for n > 2 "I cannot solve for x".

That seems like quite an odd way of looking at it.

The problem can be rephrased as: find integers a, b, c, and n > 2 such that aⁿ+bⁿ=cⁿ.

 

[Tyrion101]

Interesting, I guess I'll continue slugging through the basics, and hopefully get to where I don't need to use "find x" as much as I am right now. Thanks for the replies, it helped a lot. I think that's why I keep zoning out in math when I keep trying to go back to it, is it seems mostly the same to me.

 

[voko]

The problem can be rephrased as: find integers a, b, c, and n > 2 such that aⁿ+bⁿ=cⁿ.

Any theorem is essentially either (\exists x \in X' \subset X) \textbf{P}(x) or (\forall x \in X' \subset X) \textbf{P}(x) which can be regarded as solutions to a problem "find x in X such that P(x) holds".

 

[xodin]

Interesting, I guess I'll continue slugging through the basics, and hopefully get to where I don't need to use "find x" as much as I am right now. Thanks for the replies, it helped a lot. I think that's why I keep zoning out in math when I keep trying to go back to it, is it seems mostly the same to me.

Math is about as cumulative as it gets. Best to learn to enjoy the repetition and master the subject at hand, as you will use every single aspect of it down the road. Maybe not in your next class, or the class after that, or the class after that, but then you'll run into classes that expect you to remember exactly how to do everything you covered in algebra, geometry, trig, etc; and in many cases, expect you to know or learn on the fly the stuff that wasn't necessarily covered in detail before, such as obscure trigonometric identities, etc. Plus, as an aside, don't expect to be using calculators in advanced math, or to be able to use formula sheets on tests, but if you pay attention, do the coursework, and learn the material as you go, you won't need them.

 

[DonAntonio]

The problem can be rephrased as: find integers a, b, c, and n > 2 such that aⁿ+bⁿ=cⁿ.

Psst, that's a trivial question with a trivial answer:
a=0\,\,or\,\,b=0\,\,or\,\, c=0\,\,,\,\,\forall\,n\in\Bbb N
are (infinite) solutions.

DonAntonio

Ps. Pheew! Good it wasn't required \,abc\neq 0 \, , otherwise I'd have to Wiles-Taylor this question out!

 

[chiro]

Mathematics is about three things: transformations, constraints, and representation.

Representation is how you define something. Transformation is how you change something and a constraint is basically a kind of assumption that you use to make problems tractable and also for actually defining something (if you can't define something or it's too unconstrained, you won't be able to analyze it).

The above hold for every single area of mathematics without exception, and this is pretty much what it's all about.

With your problems of "finding x" your goal is to "find x" and in the process you start off with all your initial conditions and then you are "transforming your way" to a solution. You may make extra assumptions along the way or approximations but the idea is the same.

We do this in proofs as well: again it's invariant with respect to the field of mathematics.

 

[I like Serena]

Ps. Pheew! Good it wasn't required \,abc\neq 0 \, , otherwise I'd have to Wiles-Taylor this question out!

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? :shy:

 

[DonAntonio]

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? :shy:

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

 

[SteveL27]

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.

It's all about finding x!

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

 

[chill_factor]

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

 

[Mute]

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

A scientist might claim it's the other way around. ;)

 

[pwsnafu]

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? :shy:

0 is an integer. The set of integers is $\Bbb Z$.

 

[smize]

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.

 

[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.

 

[SteveL27]

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.

 

[DonAntonio]

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.

 

[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.

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.

 

[voko]

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?

 

[DonAntonio]

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.

 

[DonAntonio]

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

 

[voko]

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.

 

[I like Serena]

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.

 

[micromass]

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.

The point is that nobody phrases it like that. So I doubt this is what the OP had in mind.

 

[Unstable]

My observations about „higher“ math in PhD projects and research are that the topics strongly changed from the typical pure math to more applied math in the last years. At least in Germany, Universities force their departments to collect money from industry. That means you have to do something of interest for the application. Ten years ago in my university the ratio between pure and applied mathematician was 50:50. Today, I guess that more than 75% are applied. To get to the point, most projects are actually about „Find x“ (of course in most cases approximately) and I like this trend.

 

[I like Serena]

The point is that nobody phrases it like that. So I doubt this is what the OP had in mind.

I find the discussion an interesting one.
The OP appears to be already satisfied.
I am not.

Don Antonio's argument that you can/should not apply it to the real world is compelling and convincing.
Indeed that just makes no sense.
Good examples.

However... does it apply to math?
I'm not sure...
The way I rephrased the problem is nothing more than the application of the definition of abelian.
That nobody usually phrases it like that, only means that mathematicians like to put things in a short hand and define terms for them.

 

[DonAntonio]

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.

No, it can't, and if you insist in "theorem", then piece of cake:

Th. 1: If a function is derivable at some point then it is continuous at that point

Th. 2: A continuous function defined on a closed, bounded interval is uniformly continuous there

Th. 3: A subgroup of a group is normal iff it is the kernel of a group homomorphism.

Th. 4: Any rectangle whose diagonals are perpendicular to each other is a square...

Now, you can try stretches like saying: in Th. 3 , for a subgroup of a group "find x=group homomorphism" s.t. its kernel is

that subgroup...or show that such doesn't exist", but I claim this bastardizes both the intention of what actual mathematics is and

also the deep meaning of it within some given realm.

"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.

Well, I think that by now it should be crystal clear I dislike that interpretation as I think it is a bastardizing one, isn't it? And my

reason for disliking it is because, as I already explained in length, I think it is not only deeply flawed but also it is deeply misleading.

And as "valid": this seems to depend on the eyes of the beholder. For me it is completely invalid.

DonAntonio

 

[DonAntonio]

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.

Please, don't anyone take this as an offense because IT IS NOT, but the above is utterly ridiculous in my eyes: do

you REALLY mean to say that for you the problem of proving that the integers' group is an abelian one reduces to "find

the pairs ...etc."?! Really? I am assuming you're a mathematician and thus you've done mathematics at least at undergraduate

level, so it is bewildering for me how can you think your interpretation above seriously and formaly is the same as the original question...

Once again, an example out of mathematics: show that god exists/doesn exists, reduces to "find all things in the

universe/metauniverse/wherever, and show that for some x, x = god...or prove there is no such x"...Really? Common!

DonAntonio

 

[voko]

No, it can't

Yes it can. Do you need help with that?

Th. 1: If a function is derivable at some point then it is continuous at that point

I assume you mean differentiable. Let P_c(f, x) be a predicate meaning "f is continuous at x", and P_d(f, x) be a predicate meaning "f is differentiable at x". They can be represented in the epsilon-delta language, but that is unimportant for the argument. Then your theorem can be stated as \forall z \in Z = \{ (f, x) : P_d(f, x) \} P_c(f, x) The other statements you gave could equally be represented in that form.

Well, I think that by now it should be crystal clear I dislike that interpretation as I think it is a bastardizing one, isn't it? And my

reason for disliking it is because, as I already explained in length, I think it is not only deeply flawed but also it is deeply misleading.

And as "valid": this seems to depend on the eyes of the beholder. For me it is completely invalid.

This is an emotional issue for you, not rational. And your posture is offensive to many a great mathematician who would approach their theories in terms of problems to be solved.

 

[micromass]

This thread has become silly, in my opinion. This isn't worth arguing.