Re: Is Intel exceptionally unsuccessful as an architecture designer?

On 02/10/2024 23:45, MitchAlsup1 wrote:

On Wed, 2 Oct 2024 7:20:47 +0000, David Brown wrote:
 

On 01/10/2024 23:09, MitchAlsup1 wrote:

On Tue, 1 Oct 2024 19:07:18 +0000, Niklas Holsti wrote:
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On 2024-10-01 21:20, MitchAlsup1 wrote:

On Tue, 1 Oct 2024 15:51:36 +0000, Thomas Koenig wrote:
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Mathematics is not a sciene under this definition, by the way.

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Indeed, Units of forward progress in Math are done with formal
proofs.

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Yes, in the end, but it is interesting that a lot of the progress in
mathematics happens thruogh the invention or intuition of /conjectures/,
which may eventually be proven correct and true, or incorrect and
needing modification.

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Mathematical conjectures have a spectrum of "solidity" often more
solid in one branch of math than in another.
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I am not entirely sure what you mean by that.
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A conjecture is a hypothesis that you have reasonable justification for
believing is true, but which is not proven to be true (then it becomes a
theorem).  Some conjectures have been confirmed empirically to a large
degree (such as the Riemann hypothesis) which is not proof, but can be
seen as strengthening the conjecture.  Others, such as the continuum
hypothesis, not only have no empirical evidence but have been proven to
be independent of our usual ZF set theory axioms - no evidence either
way can be found.

 Other conjectures had a century or more between being conjectured with
several "things they got right" before finally drifting towards a proof
or drifting towards disproof. The width of the drift is exactly the
spectrum I stated.

Okay, so that was what you meant.  Fair enough.
Many conjectures have not "drifted" significantly one way or the other - and some have "drifted" towards an expectation (or even a proof) that they are unprovable one way or the other.

 

There are also some mathematicians who have a philosophy of viewing some
kinds of proofs as "better" than others.  Some dislike "proof by
computer", and don't consider the four-colour theorem to be a proven
theorem yet.

 Over time proofs drift towards being an axiom (at least in their little
branch of math--which might not be axiomatic in other branches). others
start out proven and drift to the point there are only proven in one
or several branches of math.

That paragraph, on the other hand, makes absolutely no sense to me.
An "axiom" is something that you take as true without any kind of proof - it is how you bootstrap mathematics.  "Two sets are equal if and only if they they have the same elements" is an axiom of ZF set theory.  "Any two distinct points can be connected by a unique straight line" is an axiom of Euclidian geometry.
Axioms are what you /start/ with - proven theorems do not become axioms over time!
And if a theorem is proven, then it is proven based on a set of axioms. It is not dependent on any particular branch of mathematics.  And it does not "drift" towards being unproven.  It can happen that mistakes are found in what was previously thought to be correct proves, but that is rare and there is no "drift".  It is also certainly the case that if you change the axioms you used to prove something, then it is not (yet) proven with the new set of axioms.  Some things can be proven if the continuum hypothesis is taken as an extra axiom - other things can be proven if you take as an axiom that the continuum hypothesis is false. But those are not different branches of mathematics, and again there is no "drift" here.

 

            Others are "constructivists" - they are not happy with
merely a proof that some solution must exist, they only consider the
hypothesis properly proven when they have a construction for a solution.
  In that sense, a given conjecture may have more "solidity" in one
/school/ of mathematics than in another.

 that is what I am talking about--it is all a big multidimensional
spectrum of {proof or conjecture}

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But I don't quite see how a single conjecture could have more "solidity"
in one /branch/ of mathematics than another.  An example or two might
help.

 A conjecture/proof in ring-sum math may not work at all in
Real-Numbers. They are different branches in the space of Math.
Some proofs only work in Cartesian Multi-D spaces and fail in
manifold spaces.
 

I think you are very confused here.
If I use ZF set theory axioms to develop Peano arithmetic and prove the theorem that for all x, y in N, x.y = y.x (i.e., that multiplication of natural numbers is commutative) then that theorem is proven correct from those axioms.  If I later define matrices and discover that matrix multiplication is non-commutative, that does not disprove my earlier theorem or show that it "works in some branches of maths but not others".
Mathematical conjectures and their proofs (if one exists) have two parts - a set of pre-conditions and a result.  The pre-conditions generally contain an implied part (a standard set of axioms) and an explicit part (such as "for all x in R, x > 0" or whatever).  The result is the bit that we conjecture is true, or have proven to be true, from the pre-conditions.  The conjecture or theorem says absolutely nothing about any other situation than when the pre-conditions hold.  It is not the case that the theorem fails in other situations - it simply has no relevance and it makes no sense to ask if it is true or not if the pre-conditions are not met.  It may be that you could formulate a similar conjecture with other pre-conditions that apply elsewhere, and that this new conjecture may be true or false, but it is a /new/ conjecture.