Is all math essentially: Solve for x?

[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.