Something about Apologetics.

[Abracadabra]

Zeno's claim is that you can't perform an infinite number of finite steps. And the calculus limit doesn't challenge that, nor does it show that Zeno was wrong.

How does a methodology for describing infinite sequences/sums/etc. not challenge the claim that you can't perform an infinite number of finite steps?

It's right there in the methodology itself.

What do you need to show in order to prove that a calculus "limit" exists?

Do you need to show that it's possible to complete an infinite number of steps?

No. Not at all. In fact, if that's what the calculus limit was saying then all the calculus limit would be proclaiming is that infinity is finite.

All you need to do to satisfy the formal calculus definition for a limit is to show the existence of trends and/or boundaries. And whether or not those trends and/or boundaries exist or not.

If you can show that certain trends exist, and they are unbounded, then you have satisfied the definition of the limit and therefore you can say that the limit "exists".

Moreover, when you have satisfied this definition and have proven the existence of the "limit" does that prove that this actual point must exist?

No it doesn't even prove that. You can easily have a function that is not include a given point. Yet, you can still prove that this undefined (and therefore nonexistent point) can still satisfy the calculus limit, and be said to 'exist' as a 'limit'. But that in no way asserts or proves that the actual point must exist. That's simply not what the calculus concept of limit is saying.

Given an infinite sum or other process, the calculus limit simply states that if you can show specific properties of trends and boundary conditions, then you are premitted to say that "If this process could be completed it would arrive at this particular target called the Limit".

But the neither the methodology used to arrive at that conclusion, nor the formal definition of the calculus limit itself make any statements requiring that this process can actually be completed, or that it should be able to be completed.

In short, the calculus limit does not assert that any infinite process is completable or even should be completable.

In fact, if it were actually saying that (which it clearly isn't) then it would be proclaiming that a finite process can indeed be completed and therefore it would have proved that infinite is finite.

The calculus limit does not do that, nor does it even imply that such a notion should be possible.

Just because it allows you to say what the sum of an infinite addition would be IF you could supposedly complete the sum, doesn't mean that it's saying that such a summation process could actually be completed.

This is a gross misunderstanding of the calculus limit.

The formal epsilon-delta definition of the calculus limit cannot be used to support any such conclusions.

That's not what the calculus limit is saying.

So the calculus limit does not having anything at all to do with Zeno.

Zeno holds that you could never complete an infinite number of tasks.

The calculus limit does not challenge this.

The calculus limit has nothing to do with Zeno's objections concerning the idea of being able to actually complete an infinite number of tasks. They simply have nothing to do with each other at all.

Yet, unfortunately many mathematicians have been erroneously taught that the calculus limit solves Zeno's concerns. I'm certain that it doesn't and if Zeno were alive today I'm sure he would object as well.

The calculus limit is addressing a totally different concept.

[Categories+Sheaves]

The calculus limit has nothing to do with Zeno's objections concerning the idea of being able to actually complete an infinite number of tasks. They simply have nothing to do with each other at all.

When a mathematician asserts that the real numbers are isomorphic to the ring of cauchy sequences in the rational numbers modulo said sequences that converge to zero, how does this fail to assert that we can treat the product of an infinite number of tasks as a completed object?

[Abracadabra]

The calculus limit has nothing to do with Zeno's objections concerning the idea of being able to actually complete an infinite number of tasks. They simply have nothing to do with each other at all.

When a mathematician asserts that the real numbers are isomorphic to the ring of cauchy sequences in the rational numbers modulo said sequences that converge to zero, how does this fail to assert that we can treat the product of an infinite number of tasks as a completed object?

It doesn't.

But what do Cauchy sequences or even the concept of 'real numbers', have anything at all to do with Zeno's concerns?

The very notion of 'real numbers' is a man-made mathematical concept that has nothing at all to do with what Zeno was talking about.

You need to realize that the very notion of a "number line" that itself is assumed to be a "continuum" is the product of the imagination of mathematicians who themselves have arbitrarily assumed that reality can be correctly described as a continuum. So they are already assuming a continuum by their own artificial definitions.

I personally don't even agree with human mathematics in the details. Much of it is nothing but man-made whims.

In fact, I personally believe that if aliens came here from another planet they may very well have a system of mathematics that is not based on the idea of a 'continuous number line' like we do.

On the contrary I would expect that since they are advanced enough to be capable of interstellar travel they would probably have a mathematics based on a quantum number line that is more in harmony with the physical nature of reality.

I personally believe that humans will eventually move beyond their current man-made mathematical formalism and move to a mathematics that's more closely aligned with physical reality.

I also suspect that this will most likely happen fairly soon. Probably within the next century or so. Although mathematicians certainly aren't moving in that direction at the present time.

[Categories+Sheaves]

But what do Cauchy sequences or even the concept of 'real numbers', have anything at all to do with Zeno's concerns?

I think this guy explained that connection pretty well:

To begin with Zeno was simply arguing that IF the world is a continuum, then motion would not be possible.

Unless, of course, your definition of 'continuum' is not related to the mathematical one. In which case, what are you taking 'continuum' to mean?

On the contrary I would expect that since they (hypothetical aliens) are advanced enough to be capable of interstellar travel they would probably have a mathematics based on a quantum number line that is more in harmony with the physical nature of reality.

Quantum number line? Do elaborate.

[Abracadabra]

Unless, of course, your definition of 'continuum' is not related to the mathematical one. In which case, what are you taking 'continuum' to mean?

There's only one concept of a continuum as far as I'm aware. It's the concept that things are continues rather than discrete. In fact, this was the very concepts that were being argued in ancient Greece.

Is reality a continuum or does it break down at some level and is actually made of discreet discontinuous quanta?

That was the question.

Zeno argued for the quantized reality as opposed to a continuum.

The people who favored the continuum ultimately won in terms of shaping mathematical thinking and this is why we currently have a mathematics that based on the idea of a continuum.

However, science and physics have shown us that reality does not think it's continuous. On the contrary, reality demands quantization, and thus the science and physics communities have acknowledged that we live in a quantum universe.

Yet, mathematicians still believe they are living in a continuum.

Quantum number line? Do elaborate.

Rational numbers are quantized.

We invented the 'reals' and forced them to become a part of the number. In doing so we had no choice but to demand that the number line itself must be a continuum.

This is totally unnecessary. The 'real numbers' have no physical existence in reality. That's an illusion created by incorrect abstract thinking.

In other words, there does not exist in physical reality a physical 'quantity' that can be described by a 'real number'. No such absolute quantization is possible in our physical universe.

I realize that you are going to give the standard response and point to the irrationality of the diagonals of squares, and the relationship of Pi, etc.

But those are not absolute quantities. On the contrary those are relative relationships between quantities that are actually dependent upon a very non-ridge and non-static fabric of spacetime.

In truth, if you could physically measure the diagonal of a square or the diameter or circumference of a physical circle down to the quantum level you'd soon discover that it's either here or there, but can't be in between.

In other words, by the time you got down to the quantum level your theoretical continuum would break down and the quanta of reality would become apparent.

The imagined 'irrationality' that we are abstractly placing on these relative measurements is not "Real". It's not based on an infinitely divisible continuum. That was an incorrect assumption to begin with.

We have no need for "Real Numbers" in an absolute sense.

And that is the important key concept to understand. The only time we require "Real Numbers" is in the case of irrationality, which only arises in self-referenced relative situations. It's not an absolute quantitative property of the universe like our current mathematical formalism is attempting to treat it.

We have no need to support these notions of "real numbers" as though they must be an integral part of our idea of absolute quantity.

In fact, by doing that, we are actually loosing site of important information.

For example, mathematicians are never going to recognize that all irrational relationship arise from self-referenced relative situations.

The reason they are never going to recognize this is because they have already accepted irrational 'numbers' as completely independent absolute ideas of quantity.

That's a mistake. No such absolute quantities exist in our physical universe.

Real number (basically irrational numbers truly), are always (and may I repeat always) due to relative self-referenced situations.

They have nothing to do with any idea of an 'absolute' quantity and our attempt to force them to become such concepts is misplaced.

We don't need anything on the number line but rational numbers, and they are discrete (i.e. quantized).

So until the mathematical community recognizes the difference between absolute quantities and relative self-referenced quantities, our mathematics will remain flawed (i.e. incorrect with respect to the true quantitative nature of reality)

We live in a quantum universe. Not a continuous one like had been thought by the ancient Greeks.

Yet, our mathematics is still based on these ancient notions that it must support a continuum.

That's simply the wrong picture of reality. And we even know this to be true from science. Yet we still cling to these ancient incorrect ideas that we live in a continuum and that it makes sense to build a mathematical formalism based on those ancient and incorrect ideals.

We need to backtrack and rethink some of this stuff.

In fact, it's not nearly as bad as it sounds. We really only need to backtrack a few hundred years actually. Just to the dawn of the formalization of Set Theory. That's were these erroneous ideas became formalized. There we some bad decisions made at the beginning of the 20th century and we need to go back and readdress those.

Calculus itself wouldn't change much at all actually. There's nothing wrong with calculus. It's various axioms of set theory that's really at the base of the problem.

But I'm sure humans will sort it all out eventually.

It's just a matter of time.

Unfortunately though most mathematicians are quite happy just accepting things the way they are. They don't see where there's a problem.

In fact many of them take the stance that it's not even the job of mathematics to correctly reflect the real world. They view it as a totally abstract product of pure human thought.

And of course they are perfectly correct! That is indeed precisely what mathematics has become. Mathematicians are so lost in pure thought they they don't even care whether mathematics matches up with the true quantitative nature of reality. That's become a totally unimportant to them.

And thus with that frame of mind they are naturally going to go off in la-la land making up mathematics on their own whim with total disregard to whether or not is means anything. As long as it seems to be "self-consistent" with respect to its axioms that's all they care about.

But even a whim that is 'self-consistent' is still just a whim.

If the axioms don't match reality, then mathematics has nothing to do with reality. And that's the bottom line right there.

If we want a mathematics that can help take us to the stars, it had damn well better be correct and true with respect to describing the true nature of the quantitative essence of reality.

Whims won't make it to the stars no matter how logically consistent they are with respect to arbitrary man-made axioms.

[Nine]

I have been holding this back.. Can't stop myself!!





That's better

<3 you Abracadabra

[LastPoet]

You should stay away of math Abra, the certainty of mathematics doesn't combine well with your 'spirits'.

[Categories+Sheaves]

Is reality a continuum or does it break down at some level and is actually made of discreet discontinuous quanta?

That was the question.

I thought we were talking about Zeno's paradoxes?

Our universe does seem to behave discretely, but Zeno's paradoxes are not proof of this of this fact. At least that's what I was arguing.

The people who favored the continuum ultimately won in terms of shaping mathematical thinking and this is why we currently have a mathematics that based on the idea of a continuum.

It's not like discrete math went extinct or anything. If anything, our physics is based on real analysis. There's still plenty of mathematicians doing research in Algebra, Combinatorics, etc.

We don't need anything on the number line but rational numbers, and they are discrete (i.e. quantized).

You probably know this already, but quantization also means our universe doesn't require the entirety of the rational numbers either.

We have no need to support these notions of "real numbers" as though they must be an integral part of our idea of absolute quantity.

This is absolutely true. We support these notions of "real numbers" because modern analysis is incredibly useful. Did I mention that it forms the backbone of QM?

According to a theory that runs on infinite-dimensional spaces of complex functions, real-world quantities are discrete. I guess that means it's time to get rid of those whimsical abstractions called real numbers!

Can you see why I think this is unreasonable?

I understand that you believe we'll get more mileage out of a system based on 'absolute' quantities. I do not understand why you think this system will afford us some problem-solving techniques that the current implementations of mathematics doesn't. A paradigm shift in mathematics would be awesome. But that sort of thing only happens when you find a system that's better than the current one.

It's various axioms of set theory that's really at the base of the problem.

I'd like you to expand on that point... but you are aware that the machinery of QM relies on that stuff too, right?

Axiom of choice? Clearly a worthless abstraction. It's not like physicists needed their hilbert spaces to have orthonormal bases or anything...

...And thus with that frame of mind they are naturally going to go off in la-la land making up mathematics on their own whim with total disregard to whether or not is means anything.

Reminds me of a guy I know.

[Doubting Thomas]

Ah, I finally remembered the definition I came up with. Or heard someone else say, or something. But the best definition of apologetics is the art of coating a dog turd in sugar and trying to sell it as a donut.

[Abracadabra]

I thought we were talking about Zeno's paradoxes?

Our universe does seem to behave discretely, but Zeno's paradoxes are not proof of this of this fact. At least that's what I was arguing.

I absolutely agree with what you are saying here. Zeno didn't "prove" anything. Nor did I claim that he did.

However, what he did do is offer a reasonable hypothesis. A hypothesis that has subsequently been observed to be true via the science of Quantum Physics.

All I'm saying is that Zeno's hypothesis was correct. I'm not saying that he had necessarily 'proved' it. But I feel that he made very good arguments that have never been refuted.

And, as I say, modern proclamations that the calculus limit answers Zeno's concerns are simply false. It does no such thing.

Zeno was right.

And that's all I'm saying.

The people who favored the continuum ultimately won in terms of shaping mathematical thinking and this is why we currently have a mathematics that based on the idea of a continuum.

It's not like discrete math went extinct or anything. If anything, our physics is based on real analysis. There's still plenty of mathematicians doing research in Algebra, Combinatorics, etc.

Oh absolutely!

Our modern "mathematics" is a truly ill-defined field in the larger picture. By that I mean that there are many different forms of analysis that are placed under the umbrella called "mathematics".

Hells bells, even Boolean algebra is considered to be "mathematics" yet Boolean algebra has nothing at all to do with ideas of 'quantity' as in normal mathematics. It's a totally different type of logical analysis having it's own rules of operations that are totally separate from the rules of numerical arithmetic.

In arithmetic there are operations like addition, subtraction, multiplication, division, etc. In Boolean algebra there are operations like AND, OR, NOT, XOR, and so on. These are totally different concepts from the normal numerical concepts of quantity. Yet we lump this into "mathematics".

The very term "Mathematics" has become nothing more than an umbrella-term used by universities as a box to shove anything into that appears to have a logical structure.

That itself is a very unwise thing to do.

Why is it unwise? Because by doing that we tend to loose sight of what's truly important in things like the study of "quantity" or "quantitative properties" that might be associated with our physical reality.

In other words, "mathematics" has become a hodgepodge collection of any sort of logical structures, and therefore it has lost sight in what's truly important or cardinally foundational to any one of them.

We'd be far better off keeping these totally different types of logical systems separate from each other.

We don't need anything on the number line but rational numbers, and they are discrete (i.e. quantized).

You probably know this already, but quantization also means our universe doesn't require the entirety of the rational numbers either.

I never said it did.

In fact, IMHO, it's utterly absurd to even speak of "The entirety of the rational numbers"

What would that even mean? The rational numbers are infinite as an abstract concept. To speak of "The entirety of them" would be to imply that you could someone include all of them. But that is Georg Cantor's mistake of tying to treat the infinite as though it can be treated as a finite thing.

All I'm saying is that "all you need" to explain the real world are rational numbers (in an absolute cardinal sense). But that doesn't mean that you need "all of them" to do this.

In fact, as you point out, in a quantized universe not only wouldn't you need them, but neither could they all exist in terms of 'real' quantitative properties.

Absolutely!

You're right on the money there.

We have no need to support these notions of "real numbers" as though they must be an integral part of our idea of absolute quantity.

This is absolutely true. We support these notions of "real numbers" because modern analysis is incredibly useful. Did I mention that it forms the backbone of QM?

According to a theory that runs on infinite-dimensional spaces of complex functions, real-world quantities are discrete. I guess that means it's time to get rid of those whimsical abstractions called real numbers!

Can you see why I think this is unreasonable?

Sure I can. But this is only because you're not truly understand where I'm coming from. I have no problem with abstractions and things like infinite-dimensional spaces, complex functions, etc.

And neither am I saying that we need to get rid of whimsical abstracted called "real numbers". On the contrary those concepts are indeed required.

What I'm actually saying is that we are simply treating them incorrectly.

I'm not saying that we need to do away with them altogether.

Pi will be Pi forever. The square root of two will be the square root of two forever. And both of those "numbers" will forevermore incommensurable in terms of rational numbers. As will be "e" of the natural logarithms, and phi of Fibonacci numbers. All of those irrational relationships will remain valid observations.

However, where the mistake is made is in demanding that these concepts be made into 'cardinal quantities' and be incorporated into a number line, etc.

That's where the mistake is made.

These irrational relationships arise from specific self-referenced situations and are not the result of cardinal properties of quantity. Especially in terms of sets or collections of "individual" cardinal objects.

So it's a mistake to try to force them to become part of the definition of cardinality. Which is what mathematicians have been doing since the early 20th century.

It's a real mistake, and one that hasn't gone unrecognized by other mathematicians. Henri Poincare had this to say about Georg Cantor's ideas of set theory (which form the cardinal basis of modern set theory)

It goes something like this, although this not an exact quote,...

"Georg Cantor's set theory based on nothing (i.e. the empty set) and producing transfinite numbers (infinities of different cardinal sizes) is a disease which future mathematicians will eventually need to be cured from"

I am totally in agreement with Henri Poincare's observations.

And he's not the only prominent mathematician who felt this way at the time. Kronecker also rejected Cantor's nonsense and stated the following:

"God gave us the integers, the rest is the work of men".

Hopefully Kronecker was speaking of 'God' in the same Albert Einstein thought of 'God'. But I also agree with the words of Kronecker. Especially in terms of a cardinal definition of number as an idea of absolute quantity.

There's simply no need to be destroying that notion with made up ideas of so-called "real cardinal numbers", that ultimately force us into the obvious paradox of being faced with infinities of different cardinal sizes.

These are not only unnecessary ideas, but they are ultimate "wrong".

They also blind us to the true nature of the relative essence of irrational relationships and the FACT that they are produced by self-referenced situations. The mathematical community doesn't even know this FACT.

They can't see it because they have indeed accepted irrational "numbers" in terms of cardinal ideas of quantity. Thus they aren't even looking for another explanation.

I understand that you believe we'll get more mileage out of a system based on 'absolute' quantities. I do not understand why you think this system will afford us some problem-solving techniques that the current implementations of mathematics doesn't. A paradigm shift in mathematics would be awesome. But that sort of thing only happens when you find a system that's better than the current one.

I can not only offer a better system than the current one, but I can even point to precisely what changes need to be made, and why they need to be made.

Precisely how that will help in a practical matter in terms of science I cannot predict. However, surely a correct mathematical formalism would be more effective than the false one we are currently using.

One immediate thing I can point to is that the corrected system shows clearly where irrational relationship come from and why they should not be thought of in terms of "cardinal" absolute quantities.

So it makes a difference already. Plus the corrected system would also not require multiple sized infinities which is another bogus idea that we don't need.

Something is either finite or infinite and that's that. The idea that something could be more infinite than something else is a bogus and unfruitful idea.

It's various axioms of set theory that's really at the base of the problem.

I'd like you to expand on that point... but you are aware that the machinery of QM relies on that stuff too, right?

Sure I'm aware of that. And this is precisely why it should be rock solid and true to reality.

I'm not saying that we should 'trash' set theory altogether. All I'm saying is that there have been mistakes made at the foundation of set theory. In fact, those errors have been introduced basically by one man - Georg Cantor.

Change those and let set theory evolve from the new foundation. They we'll have a better set theory.

In fact, if we make the changes I propose (which have also been proposed by other mathematicians even back in the days when set theory was starting out), then another great feature of this is that Kurt Godel's incompleteness theorem would also no longer apply to mathematical formalism.

The reason for that is quite involved, but that too is based on a concept of self-reference. Cantor's set theory is indeed a self-referenced system. The change I propose would produce a system that it not self-referenced.

Thus Godel's incompleteness theorem would not apply to my set theory.

But that's a whole other story.

Axiom of choice? Clearly a worthless abstraction. It's not like physicists needed their hilbert spaces to have orthonormal bases or anything...

I think you're jumping to conclusions that may not have anything at all to do with the issues that I'm attempting to address.

I'm not saying that abstraction itself is the culprit. I'm quite sure that my ideas qualify as being 'abstract' as well. But then again, that depends on what you're defining as "abstract"

What does abstract even mean?

Does it mean vague or ill-defined?
I don't think mathematicians would be bragging about mathematics being abstract if that meant that mathematics is vague or ill-defined.

Does it mean non-tangible? (i.e. having no relation to the physical world)
Well, again such a thing should not be important to mathematics. Why would mathematics brag about mathematics not relating to anything real?

However, if abstract simply means - Applicable to many cases,...
Then my ideas are as abstract as it gets.
And IMHO, this is indeed the type of abstraction that is important to mathematics.

So when we speak about 'abstract concepts' we really need to look closely at what we mean by that.

Is we simply mean, whimsical, vague or having no application to reality, then we need to question why we would even be interested in such concepts.

On the other hand, if we mean ideas that have many applications and can also be applied to correctly describing the quantitative physical properties of our universe, then I'm all for it.

...And thus with that frame of mind they are naturally going to go off in la-la land making up mathematics on their own whim with total disregard to whether or not is means anything.

Reminds me of a guy I know.

Are you sure it doesn't remind you of a guy that you truly do not know at all?