It still requires will power. It may take a long time to convince people that a particular habit is bad, but once it is recognized as bad, without the will power to stop they will continue to do it. — Metaphysician Undercover
It is not enough to point it as bad, with mathematics you must demonstrate an alternative system with superior utility, something that is better. Once a system with superior utility is at hand, the exchange would be immediate, you won't need any will power. People will readily exchange older cars for new more efficient ones if they can afford to. Its a pragmatic argument. — Zuhair
I just want to give an example of a sentence that is highly related to the finite mathematics, that can find a solution in a system that speaks of infinite objects that mathematicians seems to agree upon. That of Fermat's last theorem! This can be solved in ZFC. It's not yet know if it can be solved in PA. However the theorem is clearly about arithmetic, and its formulated in the language of PA, so it is not essentially about any infinite object. But a theory speaking about infinite objects (i.e. ZFC) can prove it. Now I'm not claiming here that ZFC had contributed to the argument of the proof of that theory, certainly not. But seeing that it is provable in ZFC and yet not known to be provable in PA yet, speaks a lot of that issue. — Zuhair
However, I would say that any proof which utilizes "infinite", or "infinity", is not a sound proof. — Metaphysician Undercover
No! not always, if the proof is carried in a FINITE fragment of ZFC, and the proved statement is an arithmetical statement, then this is already known to be SOUND, i.e. any finite fragment of ZFC (even though it speaks about infinite sets) if it proves an arithmetical statement, then that arithmetical statement is part of TRUE arithmetic, i.e. it conforms to a proof that only relies on finite objects. — Zuhair
Not only that! It is expected after knowing Wiley's proof of Fermat's Last Theorem (which he actually did it in a theory even stronger than ZFC! — Zuhair
So a theory basically about the infinite did help us understand provability within a theory about the finite, a kind of a detour though it to simplify matters! — Zuhair
That a conclusion from a theory with unsound premises happens to be consistent, or "the same" as a conclusion from a theory with sound premises, might be completely coincidental. You seem to be forgetting about all the wasted time spent using that theory with unsound premises to create conclusions which are inconsistent with the sound theory, to focus on one conclusion which coincidentally happens to be consistent, in an attempt to justify use of the unsound theory. — Metaphysician Undercover
As I stated earlier, every time that "the infinite", "infinity", or the mitigated "infinitesimal", occurs in a mathematical application, this can be judged as a bad effect of the mathematical habit. — Metaphysician Undercover
OK, might want to jettison calculus then . . .and all the technology we use as a result. — John Gill
You seem to be forgetting about all the wasted time spent using that theory with unsound premises to create conclusions which are inconsistent with the sound theory — Metaphysician Undercover
Let's go back to the beginning.
a = a + 1
1) a is NOTHING — TheMadFool
Context means everything:
(1) a=a+1 (no finite solution. In complex analysis a would be the point at infinity - corresponding to an actual point at the north pole of the Riemann sphere)
(2) a=0
for k=1 to 100
a=a+1
next
(now what is a?)
I'm still mulling over "bad habits" in math. Sloppiness; jumping over points in a proof assuming they are true; assuming a hypothesis and then proving it; muddling a proof so badly other mathematicians can't verify it; etc. Using infinity or infinitesimals are the least of our concerns. :nerd: — John Gill
Name me ONE conclusion that ZFC proved about arithmetic that is not sound? — Zuhair
I'm still mulling over "bad habits" in math. Sloppiness; jumping over points in a proof assuming they are true; assuming a hypothesis and then proving it; muddling a proof so badly other mathematicians can't verify it; etc. Using infinity or infinitesimals are the least of our concerns. — John Gill
I don't need to name any, they are all unsound. We've discussed the fact that the axioms lack truth, in how they describe objects. The axioms are the premises, and soundness requires true premises. The premises are not true, therefore the conclusions are not sound — Metaphysician Undercover
it requires good metaphysics to determine the truth about axioms — Metaphysician Undercover
I never admitted that they are not sound. They are indeed sound of what they are describing in the platonic sense. And if platonic sense proves to be indispensable for discovering our reality, by then this would prove it to be sound. So the question of soundness of those axioms and its relation to application is still unsettled. — Zuhair
But if they were unsound as you claim, then they must bear wrong theorems, i.e. we need to see MANY arithmetical consequences of those theories that violate true arithmetic. — Zuhair
All I can say is that practicing mathematicians usually avoid these discussions unless they are in these sub-disciplines. — John Gill
The undesirable consequences only become apparent in application, because the premises concerning the nature of an object are inconsistent with what an object really is. You can see these undesirable consequences in the particles of particle physics. — Metaphysician Undercover
Don't yout think it is a sort of problem, that mathematicians would avoid discussions concerning the truth or falsity of their axioms? — Metaphysician Undercover
EXAMPLES? — Zuhair
Non-standard analysis lives within a mathematical model that, to the best of my knowledge, is consistent. — John Gill
It assumes (axioms) the existence of infinitesimals and infinity with symbols representing them and rules for manipulating these symbols. Can you determine the "truth" or "falsity" of these axioms? (no fair resorting to "manifest") — John Gill
ou can see these undesirable consequences in the particles of particle physics — Metaphysician Undercover
It's called the uncertainty principle. — Metaphysician Undercover
I thought that uncertainty principle had nothing to do with the mathematics involved, it has something to do with inability of have complete form of measurement which is due to the nature of the objects studied and not to the mathematics involved in them. Not sure, really. Can you clarify the picture to me? — Zuhair
I thought the source of the problem is our "physical" means of measurement not the mathematical side of it. — Zuhair
I wanted to know what are your objections to the uncertainty principle? and why you think it is the mathematics involved in it that are the source of the problem? — Zuhair
It is not an issue of the human capacity to observe, because we already extend that capacity with instruments. Nor is it an issue of the "physical means of measurement", because we create and produce these, the instruments for measuring, as required. Therefore we ought to consider that the problem, which is causing this limit to appear before us, is a manifestation of the principles by which we interpret the information. — Metaphysician Undercover
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