Is there a limit to mathematical progress?

[andyroo]

I'm not sure myself.

 

[phinds]

Uh ... say what?

What is your question?

 

[DaveC426913]

Vague.

 

[disregardthat]

Doubt it.

 

[mathwonk]

first you have to prove progress is cauchy.

 

[Allenman]

_t\stackrel{lim}{\rightarrow}_\inftyf(mp)?

 

[Number Nine]

first you have to prove progress is cauchy.

This is my favourite post ever on this site.

 

[Functor97]

Godels incompleteness theorem seems to say we can keep on doing Mathematics (Finding interesting relationships) for eternity.
Sadly we don't have that long : (

 

[lavinia]

I'm not sure myself.

One mathematician believed that all of the interesting questions in mathematics would be answered by the end of the 20'th century. He believed that mathematical progress was already near its end. After he made this statement many new theories were discovered.

To me, as long as the physics of Nature is not completely understood, more mathematical progress is possible.

 

[DaveC426913]

Sadly we don't have that long : (

I dunno. For me, it's goin' good so far.

 

[camel_jockey]

Godels incompleteness theorem seems to say we can keep on doing Mathematics (Finding interesting relationships) for eternity.
Sadly we don't have that long : (

Is this really true or just a loose extrapolation that somebody has made? I would very much like to see the link!

:)

 

[DaveC426913]

Is this really true or just a loose extrapolation that somebody has made? I would very much like to see the link!

:)

Just Google or Wiki Godel's Incompleteness Theorem.

 

[lavinia]

Godels incompleteness theorem seems to say we can keep on doing Mathematics (Finding interesting relationships) for eternity.
Sadly we don't have that long : (

yes but these theorems don't really add to knowledge. Its is new theories that reveal new structures and new unities that is the ultimate goal.

 

[DaveC426913]

yes but these theorems don't really add to knowledge. Its is new theories that reveal new structures and new unities that is the ultimate goal.

The point of Godel's Incompleteness theorem is that it says (oversimplying) that we can never have a truly complete theory of anything, even in principle.

The implication is that there will always be more to develop and discover.

 

[camel_jockey]

I checked out the wiki article on Gödel 1 & 2... and no I didn't see any consequences for the limitations (or lack thereof) of the extent of mathematics. Where can I find it? Is it in the wikipedia article?

Not arguing here, I just want to find it :)

 

[SteveL27]

I'm not sure myself.

Well ... there was certainly a limit to MY mathematical progress! And if each human has a limit to their mathematical progress ... and if the number of humans who ever lived or ever will live is bounded ... then we can put an upper bound on mathematical progress.

But I doubt that we've reached it yet.

 

[DaveC426913]

... and if the number of humans who ever lived or ever will live is bounded ...

Why would 'the number of humans who will ever live' be bounded? Do you know something we don't?

 

[SteveL27]

Why would 'the number of humans who will ever live' be bounded? Do you know something we don't?

I clearly said IF the universe is bounded ... THEN such and so. Not sure why an IF/THEN would be misconstrued as asserting the antecedent in this esteemed group :-)

If the future history of the universe or the future history of humanity is bounded, then the number of humans would be bounded. But of course there's no way to know whether that will turn out to be the case or not.

 

[Number Nine]

Why would 'the number of humans who will ever live' be bounded? Do you know something we don't?

The lifespan of the Earth is bounded by the lifespan of the sun, which is finite. To establish an upper bound (not necessarily the LUB) on "human quantity", we can take...
P = ce^t
...where P is the number of humans currently alive, c is some constant, t is a unit of time denoting the years since the origin of the human species (t = 0). We assume that population grows exponentially (reasonable in general, though this probably overestimates population increase). To find an upper bound on the number of humans who will ever live on Earth, take...
max(h) = \int^d_0 ce^t\,dt
...where d is the date of the complete obliteration of everything on Earth forever due to supernova and subsequent death of the Sun. It's entirely possible that humans will colonize other worlds before this time, but as t tends towards infinity, the probability of a species destroying cataclysm occurring approaches one. In probabilistic terminology, human population size is almost surely bounded.

 

[Functor97]

I checked out the wiki article on Gödel 1 & 2... and no I didn't see any consequences for the limitations (or lack thereof) of the extent of mathematics. Where can I find it? Is it in the wikipedia article?

Not arguing here, I just want to find it :)

That is funny because there is an entire topic http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems#Discussion_and_implications.

The theorems' implications are still contested, read some of Gregory Chaitin
We can keep playing with set theory and n'th order logic, so mathematicians are never going to be out of a job! (the same cannot be said of those that work on wallstreet )

I leave you with my favourite quote from Godel: "Either mathematics is too big for the human mind or the human mind is more than a machine"

 

[phinds]

The lifespan of the Earth is bounded by the lifespan of the sun, which is finite.

Nonsense. You have nothing to back this up at all (mankind's life span that is, I agree w/ you about the sun). Who knows what mankind will be doing by the time the sun expires?

 

[phinds]

I clearly said IF the universe is bounded ... THEN such and so. Not sure why an IF/THEN would be misconstrued as asserting the antecedent in this esteemed group :-)

If the future history of the universe or the future history of humanity is bounded, then the number of humans would be bounded. But of course there's no way to know whether that will turn out to be the case or not.

Steve, I certainly have to agree w/ you that your statement, when taken literally, is exactly as you describe, but there is SOMETHING about the particular gramatical construct that you used that caused both me and Dave to interpret it (incorrectly, I agree) as Dave stated. I thnk this is because politicians and other fuzzy thinkers use the construct in a way not consistent with the literal meaning of the words.

 

[DaveC426913]

I clearly said IF the universe is bounded ...

No, you didn't.

Well ... there was certainly a limit to MY mathematical progress! And if each human has a limit to their mathematical progress ... and if the number of humans who ever lived or ever will live is bounded ... then we can put an upper bound on mathematical progress.

But I doubt that we've reached it yet.

I simply asked why you thought the number of humans who will ever live would be bounded. I see it is based on the assumption that humans will never leave Earth. That's a pretty shaky assumption.

 

[disregardthat]

The point of Godel's Incompleteness theorem is that it says (oversimplying) that we can never have a truly complete theory of anything, even in principle.

The implication is that there will always be more to develop and discover.

Not necessarily I believe... Unprovable statements (such as the continuum hypothesis) is sufficient for a non-complete theory, though we hypothetically could be aware of all provable true theorems, and all unprovable statements.

But, this is sidetracking the point. All mathematics is not confined to a single axiomatic system such as ZFC. We can do mathematics outside of ZFC, and there is no telling of how many different contexts we ultimately will be working in.