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.

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.

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

There is a problem with pragmatics though, which I explained in the prior post. Utility is judge in relation to the end, as the means to that end. And the particular end, in its particular nature, is particularized, itself conformed, by the means. This leaves the means and the end in a necessary relationship of cause and effect. This particular end can only be produced by this particular means. There is really no such thing as demonstrating a better way, because a different way will bring about a different end. Therefore to judge the utility of mathematical systems it is required to judge the goals, or ends of the mathematicians.

This means that we must judge what the mathematician is trying to do with the system, rather than what the mathematician is actually doing with the system, and relate what is attempted to what is actually produced. Now, "the good" exists in potential, as the thing striven for, the desired end. When we look directly at "the good", it looses its nature as something particular, and is apprehended as a generality. For example, you might have a craving for a particular type of junk food, but when you look at that desired thing as "the good", you see that many different types of food will fulfil your hunger. As "the potential" to fulfill the need, there is always numerous options as to the particular thing to fulfill the need. So this is the first step to overcoming the deficiencies of pragmatism. We need to de-particularize the nature of 'the good" (as the end, what is sought), apprehending its true nature as something general. In understanding "the good" in this way,(i.e. the truth about "the good"), we see that pragmaticism has no bearing, it has no grounding, because utility can only be judged in relation to a particular end, and there is no such thing as the one and only end which will fulfill the need. Therefore pragmatism may be dismissed as insufficient because it provides no "system" for relating one particular end to another.

This is the way to judge a habit. We apprehend, and list all effects brought about by that action, these are "the goods", the ends produced by the action. And, we also apprehend and list the true goods, what we understand as the truth concerning the goodness or badness of these effects. The habit can be judged as bad, if there are bad effects of the action. 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.

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

I must say, I don't really know what constitutes a mathematical "proof", so this example is lost on me. However, I would say that any proof which utilizes "infinite", or "infinity", is not a sound proof. Infinity, by its very nature is unresolved, so assuming it as a premise of a mathematical proof, for the purpose of resolving an unresolved issue, only creates an illusion of resolving the issue by premising that the unresolved (infinity) has already been resolved. In other words, the unresolved is inherent within the premise, so the conclusion doesn't really provide a resolution.

I will remind you, that Pythagoras demonstrated the irrational nature of the square. The relation between two perpendicular sides of a square produces the infinite, which as I argued above is bad. This makes the square a truly impossible, or irrational figure. And, all "powers" are fundamentally derived from the square. Therefore any exponentiation is fundamentally unsound in relation to a spatial representation..

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.

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! Even though mildly so) that it can even be carried in Peano arithmetic, which is of course part of TRUE arithmetic, that's what experts on the proof say, so suppose for the sake of discussion that this happens, then that would be a clear example of a theorem of PA (a theory solely about the finite world which is HIGHLY applicable, actually the most applicable theory ever) had came to be proved first via ZFC, and that knowing that proof in that higher system served as a guide to proving it the lower reliable system. 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!

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

From my perspective, ZFC has unsound axioms concerning the nature of objects, as we discussed earlier. Therefore any proof using ZFC is unsound.

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

But your use of "stronger theory", as you explained, really means a theory with less rigorous criteria for the soundness of its premises, and is therefore actually a weaker theory, less sound.

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.

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

If you are working within a FINITE fragment of ZFC, then the result is always arithmetically SOUND (that if ZFC is consistent). It's a matter of technicality.

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.

(This thread demonstrates why one of my profs sixty years ago advised his class of grad math students not to take a course in mathematical logic.)

But it looks like everyone is having fun!

OK, might want to jettison calculus then . . .and all the technology we use as a result. — John Gill

The point is that "usefulness" as a principle to base judgement on, is misleading. The existence of "bad habits" demonstrates this fact.

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

Name me ONE conclusion that ZFC proved about arithmetic that is not sound?

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:

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

I have no idea what you're talking about. Don't worry, it's me, not you. :grin:

Name me ONE conclusion that ZFC proved about arithmetic that is not sound? — Zuhair

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. Are you using a different meaning for "sound"?

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 would say that having axioms (premises) which instead of being based in the reality of the objects we are familiar with, are based in some imaginary assumptions about imaginary objects, is far worse than sloppiness. Untrue axioms have an effect reaching into many applications, whereas sloppiness is specific to the particular application. And, sloppiness can be caught and corrected through various means of verification, whereas it requires good metaphysics to determine the truth about axioms.

Potential infinity can be expressed as:

$\lim_{n \to 0} 1/n = \infty$

Actual infinity is then:

$1/0 = UNDEFINED$

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

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. 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. Why we are not seeing any? What's available in practice witness to the contrary direction, i.e. the arithmetic sentences proved in them are true, and actually it is provable that any consistent finite fragment of ZFC is arithmetically sound. You gave a metaphysical argument against set theory which I don't totally agree with.

it requires good metaphysics to determine the truth about axioms — Metaphysician Undercover

I'm not convinced that metaphysics can determine anything and I'm not sure what is meant by the "truth" of axioms. This is an interesting discussion and a few of you are probably analytic philosophers who know much more set theory than me. All I can say is that practicing mathematicians usually avoid these discussions unless they are in these sub-disciplines. This commonly employed mathematics seems to describe most of the physical world. And, yes, the point at infinity exists, as I've described it.

But one can certainly entertain opposing ideas. That's metaphysics. :cool:

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

I thought we agreed that Platonic objects are not true objects. We assume these objects for some purpose or utility, but they do not have any real existence as objects. So if we create premises which describe Platonic objects as objects, when they really are not objects, but fictional objects, then these premises are false and therefore unsound.

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

This is not the case. "True arithmetic" is arithmetic as defined by the accepted axioms. So if unsound axioms are accepted into "true arithmetic", then we might see no such consequences if there is consistency between the unsound axioms. And of course, there is consistency in the unsound premises of Platonic objects. 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.

All I can say is that practicing mathematicians usually avoid these discussions unless they are in these sub-disciplines. — John Gill

Zuhair actually appears to be quite knowledgeable about mathematics and its axioms.

Don't yout think it is a sort of problem, that mathematicians would avoid discussions concerning the truth or falsity of their axioms? Think about other disciplines, a physician for example. Do you think a physician would be comfortable applying principles of medicine without any concern for whether the principles are true or not.

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

EXAMPLES?

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

Not at all. There are mathematicians whose expertise lie in set theory and foundations. Let them do their job. Non-standard analysis lives within a mathematical model that, to the best of my knowledge, is consistent. 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")

EXAMPLES? — Zuhair

You want examples of why conclusions drawn from false premises are unsound? Come on. Try this. The full moon is ten miles away. I can walk ten miles in four hours. Therefore I can walk to the full moon in four hours.

Non-standard analysis lives within a mathematical model that, to the best of my knowledge, is consistent. — John Gill

Consistency does not mean that the premises are true, that's the problem. A system with complete consistency, applying false premises will still give unsound conclusions.

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

Yes, we can make those judgements. We have to look at what the symbols represent, and make judgements on the reality of that If the symbols are supposed to represent objects, we can apply the law of identity, as proposed by Aristotle in his battle against sophism. We discussed this to considerable extent already in the thread. For instance, in the case of "2+2=4", I argued that if each 2 represents an object, then each 2 must represent an object which is distinct from the object represented by the other 2, or else there would be no equality with the object represented by 4.

This is the fundamental principle of counting. 1 represents an object. We add to that another object, represented by 1 (1+1), and we now have counted 2 objects. In order for there to be 2 objects in that count, each 1 must represent a different object. And, we can proceed indefinitely, to count numerous objects in this way, so long as we recognize that each time we add 1 to the count, it must represent an object distinct, and different from the objects already counted. If we count the same object over again, the count is invalidated, like in the case of the person who counts money folded over in one's fist, so that the same bills are counted twice..

ou can see these undesirable consequences in the particles of particle physics — Metaphysician Undercover

I want examples of those, I mean of the undesirable consequences in the particles of particle physics. Which known examples you are referring to?

It's called the uncertainty principle.

It's called the uncertainty principle. — Metaphysician Undercover

Oh! but that's cornerstone in Quantum mechanics, isn't it? I always hear about a lot of strange conclusions in quantum theory like all possible worlds being actuated, and all intermediate states being there, even between life and death, etc.. I don't know if these are actually of any importance. But do you think that all of those strange results are due to the nature of the platonic realm in which the mathematics of that mechanics is coined? 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?

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

Mathematics is our means for measurement. I already proposed, and you somewhat accepted, that the reason we have "infinite", or "infinity", as a feature of our measurement scale, is that this assumption gives us the capacity to measure anything. If an object appears to extend beyond our capacity for measurement, (i.e. beyond the infinite), this implies that "infinity" is not being properly applied. We cannot ever blame the nature of the object for our inability to measure it, because this is self-defeating, killing the inspiration required to devise the means for measuring it.

That's not what I've asked about in my last comment. I wanted to know how the "uncertainty principle" is the error of applying an unsound mathematical system to particle physics? 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? I thought the source of the problem is our "physical" means of measurement not the mathematical side of it. Can you elaborate on this specific issue, I mean exactly that related to the uncertainty principle.

Wiki: "In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities[1] asserting a fundamental limit to the precision with which . . ."

I thought the source of the problem is our "physical" means of measurement not the mathematical side of it. — Zuhair

Without a doubt, there is a physical limit to the human capacity to observe our world. We observe with our senses. Molecules are at the limit of what we observe with our senses. It may be argued that we taste and smell them. But any particles smaller than this are beyond our capacity to directly observe. However, we devise instruments to extend this capacity of observation. Mathematics is used in this extension, The information from the instruments is interpreted with mathematics, and assumed to be made commensurable with the things observed by the senses through the principles of application, axioms and physical theories.

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

I have no "objections" to the uncertainty principle, uncertainty is the natural product when we arrive at the limits of our capacity to understand. But the issue is, why does the limit appear here, why can't we extend our understanding further. And the answer is that we do not have the principles required to enable us to go beyond this point.

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.

Thanks, John. Also from that wiki page, I'll add "...the mathematical framework of quantum physics does not support the notion of simultaneously well-defined conjugate properties expressed by a single value."

To give Zuhair some credit though, the problem is derived from the Fourier transform which deals with our physical capacity to measure frequencies. The problem I see, is that instead of working to find the principles and axioms which will get us beyond this apparent limit, the physicists and mathematicians, accept this limit, and base their principles on assuming this limit as the real limit (special relativity), so that any real things beyond this limit will remain in the realm of uncertainty. Mathematics is inherently unlimited (infinite), but when we put a limit on physical existence, as SR does, then there is no need for mathematics to extend beyond that limit, so we accept principles which allow infinity to be reached at that limit. There is no longer any need to allow for mathematical principles to extend beyond this apparent limit, because it is assumed that the limit is real, therefore there is nothing beyond it to be measured. The result is that anything real beyond that limit cannot be measured because the mathematics has been shaped so as to disallow this, an all that is there is left as unknown.

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

Its nice to be informed of that. But my knowledge about those issues is damn sketchy. And so I have no say in such subjects. Thanks for your informative reply.

I was always under the impression that mathematics can supply us with descriptive arsenal that help us discover matters easily. Due to human nature people often don't see (i.e. overlook) what they don't seek. I think that without having descriptive account on "orbits" like those of Ellipses, Parabolas, and hyperbolas that mathematics beforehand supplied us with, it could have been very difficult to observe how the planets moves, and it would be very difficult to predict their movements. Possibly similar things might apply with the uncertainty principle. I don't know really.

It all breaks down as limits are approached:

Wiki: "The term Planck scale refers to the magnitudes of space, time, energy and other units, below which (or beyond which) the predictions of the Standard Model, quantum field theory and general relativity are no longer reconcilable, and quantum effects of gravity are expected to dominate. This region may be characterized by energies around 1.22×1019 GeV (the Planck energy), time intervals around 5.39×10−44 s (the Planck time) and lengths around 1.62×10−35 m (the Planck length). At the Planck scale, current models are not expected to be a useful guide to the cosmos, and physicists have no scientific model to suggest how the physical universe behaves. The best known example is represented by the conditions in the first 10−43 seconds of our universe after the Big Bang, approximately 13.8 billion years ago. "