Reply to: Fermat

[Churchill]

You haven't read Kuhn, then?

Yes "knowledge" accretes for a while, then there is a "paradigm shift" which changes everything.

Newton wasn't wrong, his model works at non-relativistic speeds and for relatively low levels of gravity (hence his equations are used for space travel), Einstein came along and made a more complex model that explained things for a wider set of conditions. That's accretion.

Because Quantum theory (which is a paradigm shift) and Spec. and Gen. Rel are so incompatible there's something we're not getting/ missing in either or both.
When that is explained, I suspect we're going to have to unlearn a lot of what we think we know now.

Newton wasn't wrong when he was taking the piss out of Hooke though, was he?

[Churchill]

Oy you lot. Stop hijacking my thread!



The assignment was to show that there does not exist three consecutive natural numbers whose product can be expressed as the form a^k, where a and k are also natural numbers (k>1).

So plagiarism is what's taught in school these days, eh?

[sasguru]

Each new layer of understanding very rarely invalidates the old previously accepted one because they are only accepted after holding up to intense empirical scrutiny. We don't say relativity proves Newton wrong, or that field-theory proves Einstein wrong.

You haven't read Kuhn, then?

Yes "knowledge" accretes for a while, then there is a "paradigm shift" which changes everything.

Newton wasn't wrong, his model works at non-relativistic speeds and for relatively low levels of gravity (hence his equations are used for space travel), Einstein came along and made a more complex model that explained things for a wider set of conditions. That's accretion.

Because Quantum theory (which is a paradigm shift) and Spec. and Gen. Rel are so incompatible there's something we're not getting/ missing in either or both.
When that is explained, I suspect we're going to have to unlearn a lot of what we think we know now.

[d000hg]

Our understanding of gravity has changed massively over the last few hundred years - and the announcement about inflationary universe findings the other week mark another change, but its still called gravity and in 99% of cases the old Newtonian system is still "correct".

Each new layer of understanding very rarely invalidates the old previously accepted one because they are only accepted after holding up to intense empirical scrutiny. We don't say relativity proves Newton wrong, or that field-theory proves Einstein wrong.

[sasguru]

That's wrong on two counts

1. Relativity predicts the existence of black-holes
2. They are explained (to an extent) by current theories - specifically general relativity.

Fair enough I stand corrected. But I'll wager that our "understanding" of relativity and quantum theory and black holes will have changed quite radically a few hundred years from now, if they even have those names.
That's the point I was trying to make.
A second point I was alluding to, is that there is a school of though that reckons all of this stuff is made up by humans and their very limited pattern finding apparatus and we impose these "rules" on the world, when the underlying reality is something completely different - my thoughts on that are that if that were true why would some theories have more predictive power than others?
"Facts" are not what they seem.

[d000hg]

I've always assumed they are called black holes because no light escapes.

Now there it does get more complicated and open to debate. For instance google "Hawking Radiation", but the precise behaviour of a BH is something still a subject of academic research.

[d000hg]

It must have been pretty obvious to everyone except you that doohg meant black holes were _just_ a theory at the time he was speaking of, and at that time none had yet been observed.

But any excuse to get on someone's case and start lecturing ..

(and yes, they have now been observed indirectly, to such an extent that no mainstream sane physicist doubts their existence)

Exactly Owly. He really jumped in with both feet this time.

[d000hg]

What that article shows is that Dooghie was talking total bollocks.
He claimed that black holes emerged from our theories of physics whereas it's clear that whatever they are, which we don't know yet (hence the term "black holes", rather like "black box"), they are an observed phenomena that we can't explain with our current theories.
So much for a science degree, eh?
And there you have, in a nutshell the problem with what has gone wrong with British education in the latter years.

You really ought to learn to stop before making such a fool of yourself. The existence of black holes was predicted BEFORE they were observed in nature, in fact people went looking for them based on the prediction.

Read a book why don't you?

[doodab]

I've always assumed they are called black holes because no light escapes.

[NotAllThere]

What that article shows is that Dooghie was talking total bollocks.
He claimed that black holes emerged from our theories of physics whereas it's clear that whatever they are, which we don't know yet (hence the term "black holes", rather like "black box"), they are an observed phenomena that we can't explain with our current theories.

That's wrong on two counts

1. Relativity predicts the existence of black-holes
2. They are explained (to an extent) by current theories - specifically general relativity.

[sasguru]

Firstly, sadly physicists take little notice of Stephen Hawking's pronouncements these days. The charitable interpretation is that they are deliberately provocative and contrarian, to increase sales of his pot boilers.

Also, in case you cited that article to cast doubt on the existence of black holes, might I point out that it states the following near the start:

What that article shows is that Dooghie was talking total bollocks.
He claimed that black holes emerged from our theories of physics whereas it's clear that whatever they are, which we don't know yet (hence the term "black holes", rather like "black box"), they are an observed phenomena that we can't explain with our current theories.
So much for a science degree, eh?
And there you have, in a nutshell the problem with what has gone wrong with British education in the latter years.

[OwlHoot]

Oy you lot. Stop hijacking my thread!



The assignment was to show that there does not exist three consecutive natural numbers whose product can be expressed as the form a^k, where a and k are also natural numbers (k>1).

OK, so formally there is no pair of integers x and y with (x - 1) x (x + 1) = y^k.

Since x and x^2 - 1 are coprime, that would require integers u and v with x = u^k and x^2 - 1 = v^k

In other words, it requires (u^2)^k - v^k = 1 which means u^2 and v must be distinct (or else the LHS would be zero). But if they are distinct then the minimum absolute value occurs with u^2 = v + or - 1, in which case for k > 1 the difference must be greater than 1.

[NotAllThere]

Oy you lot. Stop hijacking my thread!

What was the question, out of interest?

Seems more likely that a number theory assignment would use Fermat's Little Theorem. Are you sure that wasn't what he was talking about?

The assignment was to show that there does not exist three consecutive natural numbers whose product can be expressed as the form a^k, where a and k are also natural numbers (k>1).

[AtW]

For one of the questions on my son's number theory assignment, he's invoked Fermat's Last Theorem in the proof. I couldn't have done that when I was at university - it hadn't been proved then!

Should have invoked the 5th amendment instead...

[OwlHoot]

Jeez, you lot only demonstrate your ignorance, time and time again.
You really need to read about the philosophy of science especially Popper and Kuhn.
As we're talking on different levels and your unknown unknowns are too great, carry on as you are.

As for black holes, what are the "facts" about them?

What Hawking meant when he said 'there are no black holes' | PBS NewsHour

Firstly, sadly physicists take little notice of Stephen Hawking's pronouncements these days. The charitable interpretation is that they are deliberately provocative and contrarian, to increase sales of his pot boilers.

Also, in case you cited that article to cast doubt on the existence of black holes, might I point out that it states the following near the start:

To be clear, Hawking was not claiming that black holes don’t exist. Astronomers have been observing black holes for decades, said Joseph Polchinski, theoretical physicist at the Kavli Institute for Theoretical Physics at the University of California, Santa Barbara. ...