Can Different Types of Infinities Coexist in Mathematical Reality?

[sludger13]

I always thought the derivatives (or generally any operation with infinitesimal (infinitely large) values) belongs to the different infinity (= it is different variable). And obviously, there is no transition between two infinities (that is why is it different variable - e.g. big bang singularity problem).
I thought every time only one infinity can be intended as real, in that case other infinities are just inf. smaller (i.e. 0 is expressing all values of smaller infinity) respectively inf. larger (i.e. every real value can be expressed only as 0 in larger infinity). But with this approach I stumbled.

As I think about it, I realized that something (let's call it a reality) is always overstated to all infinities, making every single infinity REAL (simultaneously, no matter those are different infinities). Thus I can write real values (5,26,356,-14...) of different derivatives (i.e. different infinities) side by side, although they are infinitely larger each other. Or I can solve differential equations.

This illustrates the image:
https://www.physicsforums.com/attachment.php?attachmentid=70445&stc=1&d=1402254253

So I want to ask: is it true consideration? Because I doubt it a little bit. For example in the picture: dx_{1} should be infinitely smaller than the rest of real graph, as I previously thought. Real ratio of those two variables evokes for me that they belongs to the same infinity.

 

[micromass]

It seems that you are very vaguely talking about the hyperreal (or related) umber system.
Try reading this (free) calculus book http://www.math.wisc.edu/~keisler/calc.html

Also, check out our FAQ: https://www.physicsforums.com/showthread.php?t=507003

 

[pwsnafu]

I always thought the derivatives (or generally any operation with infinitesimal (infinitely large) values) belongs to the different infinity (= it is different variable).

If you want to discuss non-standard analysis you need to say so.
The standard definition of derivative does not use infinity anywhere.

And obviously, there is no transition between two infinities (that is why is it different variable - e.g. big bang singularity problem).

I have no idea what this sentence means.

I thought every time only one infinity can be intended as real,

Infinity is never a real number. Again, if you wan to talk about hyperreals, you need to be explicit. Note that even in the hyperreals "infiniticimals are real" is still a false statement.

As I think about it, I realized that something (let's call it a reality) is always overstated to all infinities, making every single infinity REAL (simultaneously, no matter those are different infinities).

I don't know if you are talking about "real" as in real numbers, or "real" as in real life. The latter is a philosophy topic and not for this forum.

Thus I can write real values (5,26,356,-14...) of different derivatives (i.e. different infinities) side by side, although they are infinitely larger each other. Or I can solve differential equations.

Again, derivatives do not use infinity.

 

[WWGD]

Some down-to-earth points that I think may allow the OP to be more concrete and more Mathematical:

One of the main differences between the Hyperreals H and Reals R is that in H you can have indefinitely-small numbers that are not 0, but this is not so in R. A similar result is true for indefinitely-large numbers, but I cannot think of a clear way of stating this. There are many different constructions of H; one I think most are familiar with is the one in model theory , building new models using ultraproducts and filters, where a hyperreal number is an equivalence class of sequences, and it is a model for the standard axioms of the Reals by the Compactness theorem in logic. For more on ultraproducts , watch the TV show: "ultraproducts: America's new supermodel"(kidding.)

 

[sludger13]

Sorry for everyone, I'm perhaps creating my own definitions.

pwsnafu:

The standard definition of derivative does not use infinity anywhere.

I don't know what do you mean. A derivative is a ratio of two infinitely small values. If it wasn't defined as infinite values, it wouldn't (reflect, express) - let's say a slope of the tangent - precisely.

Infinity is never a real number.

Why couldn't I consider any number axis as real? With real number axis, there exists infinitely small number axes everywhere on the real axis (in fact those are points on the real number axis). And the whole real number axis is a single point on the infinitely large number axis. Like this, I can consider arbitrary number axis as real (then the others are inf. small/large).
That's what I meant by "I thought every time only one infinity can be intended as real". Infinity = number axis.

I have no idea what this sentence means.

That's simple. Let's take an amount of volume. Real degree of volume (i.e. belongs to certain infinity) cannot become infinitely small (i.e. cannot move into another infinity), nor conversely (I'm referring to the beginning of big bang). As the volume is decreasing, it obviously keeps always in same infinity.
If you now consider infinitely small number axis as real, that volume again cannot become infinitely large.

I don't know if you are talking about "real" as in real numbers, or "real" as in real life.

I meant of course real numbers. The physical reality (that is the only one for me) "behaves" mathematically, all mathematical notions were observed from reality (let's say from behavior of matter in timespace). I'm thinking about the concept of reality commonly. For me there is nothing philosophical about reality.I'm thinking about this:
\frac{df(x)}{dx}= infinitesimal result? Or real result? I don't understand how INFINITESIMAL result could "be equal" to REAL slope of tangent :(

And finally, a lot of patience with me, please...

 

[HallsofIvy]

Sorry for everyone, I'm perhaps creating my own definitions.

pwsnafu:

I don't know what do you mean. A derivative is a ratio of two infinitely small values. If it wasn't defined as infinite values, it wouldn't (reflect, express) - let's say a slope of the tangent - precisely.

No, it isn't. In standard calculus, the derivative is a limit, not a ratio. As others have mentioned in "non-standard analysis" we extend to the "hyper-reals" an have "infinitesimal" and "infinite" quantities but you don't seem to be talking about that.

That's simple. Let's take an amount of volume.

Then, even in non-standard analysis, you are talking about a real number. It does not belong to any kind of "infinity".

Real degree of volume (i.e. belongs to certain infinity) cannot become infinitely small (i.e. cannot move into another infinity), nor conversely (I'm referring to the beginning of big bang).

Are you talking about mathematics or physics?

As the volume is decreasing, it obviously keeps always in same infinity.Why couldn't I consider any number axis as real? With real number axis, there exists infinitely small number axes everywhere on the real axis (in fact those are points on the real number axis). And the whole real number axis is a single point on the infinitely large number axis. With this, I can consider arbitrary number axis as real (then the others are inf. small/large).

Now I have no idea what you mean by "axis". You cannot "infinitely small number axes" on a single axis using any reasonable definition of "axis".

That's what I meant by "I thought every time only one infinity can be intended as real". Infinity = number axis.I meant of course real numbers. The physical reality (that is the only one for me) "behaves" mathematically, all mathematical notions were observed from reality (let's say from behavior of matter in timespace). I'm thinking about the concept of reality commonly. For me there is nothing philosophical about reality.

Again, you seem confused about the difference between mathematics and physics.

I'm thinking about this:
\frac{df(x)}{dx}= infinitesimal result? Or real result? I don't understand how INFINITESIMAL result could "be equal" to REAL slope of tangent :(

Then I strongly recommend that you take a Calculus course. There are Calculus texts (and, I imagine, classes) that deal with Calculus from a "non-standard", "hyper-reals" point of view but I would not recommend one of those until after you have taken a traditional "limits" based Calculus class which uses only real numbers. Just developing the hyper-reals takes a heck of a lot of deep symbolic logic.

And finally, a lot of patience with me, please...

 

[WWGD]

Sorry for everyone, I'm perhaps creating my own definitions.

pwsnafu:

I don't know what do you mean. A derivative is a ratio of two infinitely small values. If it wasn't defined as infinite values, it wouldn't (reflect, express) - let's say a slope of the tangent - precisely.Why couldn't I consider any number axis as real? With real number axis, there exists infinitely small number axes everywhere on the real axis (in fact those are points on the real number axis). And the whole real number axis is a single point on the infinitely large number axis. Like this, I can consider arbitrary number axis as real (then the others are inf. small/large).
That's what I meant by "I thought every time only one infinity can be intended as real". Infinity = number axis.That's simple. Let's take an amount of volume. Real degree of volume (i.e. belongs to certain infinity) cannot become infinitely small (i.e. cannot move into another infinity), nor conversely (I'm referring to the beginning of big bang). As the volume is decreasing, it obviously keeps always in same infinity.
If you now consider infinitely small number axis as real, that volume again cannot become infinitely large.I meant of course real numbers. The physical reality (that is the only one for me) "behaves" mathematically, all mathematical notions were observed from reality (let's say from behavior of matter in timespace). I'm thinking about the concept of reality commonly. For me there is nothing philosophical about reality.I'm thinking about this:
\frac{df(x)}{dx}= infinitesimal result? Or real result? I don't understand how INFINITESIMAL result could "be equal" to REAL slope of tangent :(

And finally, a lot of patience with me, please...

Please note that, as many have said, both Mathematics and Physics use precise definitions. There is room for discussion, and it may take years before the right definition is found, but once it is found, it is made precise. If you are not dealing here with precise definitions, you are not dealing with neither Mathematics nor Physics; nothing wrong with that, but it is difficult to have a discussion with someone if that someone is using definitions other than those usually used. Maybe you want to ask why some terms are defined the way they are defined? This is not strictly Mathematics ( it is Meta- Mathematics), but still seems within the scope of this forum.

 

[sludger13]

Again, you seem confused about the difference between mathematics and physics.

That is true. I never understood what math really is. I'm mainly trying to avoid an approach that math is something unreal. I'm trying to unify my view to "math" and "physics" to the one and only reality.

The standard definition of derivative does not use infinity anywhere.

Ultimately, I have no problem with this. As every infinitesimal value can be expressed only as 0 in \mathbb{R}, the definition with 0 is correct.
Afterwards I don't see the meaning of limit in that definition. Why isn't there just written zero changes, but is there used limit? After all, the subtraction of two, above all limits near values is always 0 in \mathbb{R}. The definition with limit is correct, but why cannot be the definition simply with 0 changes?

 

[verty]

Ultimately, I have no problem with this. As every infinitesimal value can be expressed only as 0 in \mathbb{R}, the definition with 0 is correct.
Afterwards I don't see the meaning of limit in that definition. Why isn't there just written zero changes, but is there used limit? After all, the subtraction of two, above all limits near values is always 0 in \mathbb{R}. The definition with limit is correct, but why cannot be the definition simply with 0 changes?

Think of it in this way, there are no infinitesimal values, every time you think you have an infinitesimal value, you have something different. This should start to make things clearer. If you have a sequence that decreases all the way to 0 but never gets there, the limit is 0, it's not some other type of value. Then you should find that it is less confusing.

 

[sludger13]

I need some proof that division of two infinitesimal values is a real value. That's what I can't see intuitively, but my theory is based on the opposite.

 

[micromass]

I need some proof that division of two infinitesimal values is a real value. That's what I can't see intuitively, but my theory is based on the opposite.

You have not been listening to us. In the real numbers, there are no infinitesimals! They don't exist.

If you want a theory with infinitesimals, then you will need some other number system such as the hyperreal numbers. Relevant theorems can be found in the calculus book I linked in Post 2.

 

[sludger13]

Think of it in this way, there are no infinitesimal values, every time you think you have an infinitesimal value, you have something different..

I don't understand - what means something different? If there are no infinitesimal values, I would answer that the number axis is not differentiable. Or, if I consider only \mathbb{R} number axis and there are infinitesimal values, then I may be only 0.

 

[micromass]

I don't understand - what means something different? If there are no infinitesimal values, I would answer that the number axis is not differentiable. Or, if I consider only \mathbb{R} number axis and there are infinitesimal values, then I may be only 0.

What does it even mean for a number axis to be differentiable?

 

[sludger13]

I need some proof that division of two infinitesimal values in hyperreal numbers is a real value. That's what I can't see intuitively, but my theory is based on the opposite.

 

[micromass]

I need some proof that division of two infinitesimal values in hyperreal numbers is a real value.

There is no such proof since it is not true. The division of two infinitesimals can be infinitesimal, finite or infinite. See the book I linked in Post 2 page 31.

 

[HallsofIvy]

I need some proof that division of two infinitesimal values in hyperreal numbers is a real value. That's what I can't see intuitively, but my theory is based on the opposite.

Then you need to take a course in, or read a book on, "non-standard analysis". You will NOT find "infinitesimal" in any standard text on Calculus.

 

[sludger13]

What does it even mean for a number axis to be differentiable?

That the number axis is continuous - so the limit derivation definition can be applied.

 

[micromass]

That the number axis is continuous - so the limit derivation definition can be applied.

What does it mean that the number axis is continuous?

Also, you don't need any infinitesimals in order for the limit definition of derivatives to work.

 

[sludger13]

I need some proof that division of two infinitesimal values in hyperreal numbers is a real value. That's what I can't see intuitively, but my theory is based on the opposite.

However, I thought there is such proof in \mathbb{R} - that is probably false, just because there are no infinitesimal values in \mathbb{R}. Also such operations like:
\frac{\varepsilon^{2}}{\varepsilon} = infinitesimal, because = \varepsilon

are only predicted from the behavior of real values (basically we don't know whether this relation works with infinitesimal values (and it's irrelevant) - those relations are postulated from \mathbb{R}). No problem with that \mathbb{R}^{*} definition, it is probably the easiest way. I can just define infinitesimals another way.

I can probably every value from different infinity (f(x),\frac{\mathrm{d} f(x)}{\mathrm{d} x},\frac{\mathrm{d^{2}} f(x)}{\mathrm{d} x^{2}}...) consider on different \mathbb{R} number axis. I just cannot make any comparisons of those axes (I cannot compare them as two real, equal axes). The assumption is that all values from one axis are just 0 in higher axis.
As for \frac{\varepsilon^{2}}{\varepsilon}=\varepsilon, I'm doing exactly the same. I just postulate "miraculous transition" of \varepsilon value to another number axis (let's say it's decreasing from higher axis). Then I divide real numbers. Also this theory is quite similar to \mathbb{R}^{*} (if it is not the same).
The entire graph (like in my first post) corresponds to one real axis. All single values from other lower axes are then 0 in that graph.
The advantage of this theory for me is, that I can nicely imagine the impossibility of transition from one number axis (from one infinity) to another number axis (to another infinity).

As for the slope of tangent in the graph - the problem would be solved if I defined the slope in the graph as \int (\frac{\mathrm{d} f(x_{1})}{\mathrm{d} x_{1}}=0)dx. This integral expresses every tangent slope (C(int)\in \mathbb{R} - some solution of integral expresses every tangent).

I haven't redefined much, much other stuff so far. I shall see later whether this theory would be sufficiently consistent and transparent for me.

 

[WWGD]

I need some proof that division of two infinitesimal values in hyperreal numbers is a real value. That's what I can't see intuitively, but my theory is based on the opposite.

Use the sequence definition of hyperreals, and the definition of division in it, applied to infinitesimals.

 

[pwsnafu]

That is true. I never understood what math really is. I'm mainly trying to avoid an approach that math is something unreal. I'm trying to unify my view to "math" and "physics" to the one and only reality.

There are different ways in which something can be "real" in reality. A unicorn is not real because we don't have a living specimen. The concept of unicorns is real because we can define it with words. Mathematics, like "honor" or "congress", falls in the latter.

Ultimately, I have no problem with this. As every infinitesimal value can be expressed only as 0 in \mathbb{R}, the definition with 0 is correct.

Infinitesimals are defined as numbers which satisfy $0<x<r$ for every real number $r>0$. You can't express (or represent) infinitesimals with 0 because it means $0\neq0$.

There are two ways to obtain infinitesimals

1. Add elements to the reals (hyperreal method)
2. Strengthen the axioms of the reals in order to distinguish infinitesimals (Internal Set Theory method)

In neither case does 0 represent an infinitesimal.

Afterwards I don't see the meaning of limit in that definition.

Do you even know the epsilon-delta definition of limit?

Why isn't there just written zero changes, but is there used limit?

Because they are different.

After all, the subtraction of two, above all limits near values is always 0 in \mathbb{R}.

If you mean the numerator in the definition of derivative, that is always non-zero for non-constant functions.

Serious question, when your teacher showed you the definition, didn't you do any exercises where you calculate the derivative using first principles?

The definition with limit is correct, but why cannot be the definition simply with 0 changes?

Taking the limit to zero is not the same thing as substituting in zero.

That's what I can't see intuitively, but my theory is based on the opposite.

This forum is not the place for original research.
More to the point, you have severe misunderstandings about what is going on.

If there are no infinitesimal values, I would answer that the number axis is not differentiable.

That the number axis is continuous - so the limit derivation definition can be applied.

An axis is a line which we use to orient the Cartesian coordinate system. It has nothing to do with calculus.

If you want to know why we use the reals to do calculus with, it is because the reals are Dedekind-complete (that means: any subset of the reals which is bounded above has a least upper bound).

 

[sludger13]

The concept of unicorns is real because we can define it with words.

Because there exists such an arrangement of matter that creates a statue of unicorn, or a movie with moving unicorn, or a brain structure that can somehow imagine unicorn. The concept of unicorn is real because there exists an arrangement of matter that we can define as unicorn.

Do you even know the epsilon-delta definition of limit?

Yes, I know that. I have no problem with limit derivation definition.

In neither case does 0 represent an infinitesimal... Because they are different... If you mean the numerator in the definition of derivative, that is always non-zero for non-constant functions.

Because it's defined so. I'm suggesting that in physical reality there is no meaning that e.g.:
$\varepsilon =infinitesimal,\frac{\varepsilon^{2} }{\varepsilon}=\varepsilon$

Serious question, when your teacher showed you the definition, didn't you do any exercises where you calculate the derivative using first principles?

What definition do you mean? You mean the limit derivation definition? What exercises do you mean?

If you want to know why we use the reals to do calculus with, it is because the reals are Dedekind-complete (that means: any subset of the reals which is bounded above has a least upper bound).

I thought that we use reals because real numbers have meaning in physical reality.

 

[jbriggs444]

I thought that we use reals because real numbers have meaning in physical reality.

It's just a name. The "real" numbers are not real in the ordinary sense of the word. The "natural" numbers are not natural in the ordinary sense of the word. The "imaginary" numbers are not any more imaginary than any other class of numbers. The "rational" numbers are not rational in the sense of rational thought. The "irrational" numbers are not irrational in the sense of rational thought.

The real numbers are convenient to use in our models because they have enough range and precision to match any conceivable measurement and because as a number system they are closed under the kinds of calculations that we want to perform with them.

[In practice, we may actually use IEEE floating point because we can then fit descriptions of the model numbers into a computer and because we have an entire discipline within mathematics that can deal with the quantization errors that result from this departure from the ideal models]

 

[sludger13]

The "real" numbers are not real in the ordinary sense of the word...

There is an abstract meaning (either created in my brain or even "deeper" existing in physical reality) that those apples are - what we call - two. I consider that meaning as some feeling, or intuition in my brain.

 

[micromass]

There is an abstract meaning (either created in my brain or even "deeper" existing in physical reality) that those apples are - what we call - two. I consider a meaning as some feeling, or intuition in my brain.

First of all, "two" is a natural number. One could indeed argue that those have a special significance. But the real numbers consist out of way more numbers than the natural numbers. In fact, most real numbers can't even be defined or described. They just exist to make our mathematical theory more beautiful. There is no reason to expect that those numbers have a special significance in the real world, much unlike numbers like two which are special.

Second, one could even argue that the natural numbers aren't so special. There are some "primitive" tribes which only recognize the numbers 1,2,3, many. Something like 4 and 6 are not distinguished. This is very much like animals see numbers.

Also, this is getting dangerously in the realm of philosophy, something that this not permitted in the math sections. But I will allow the discussion for now.

 

[pwsnafu]

Because there exists such an arrangement of matter that creates a statue of unicorn, or a movie with moving unicorn, or a brain structure that can somehow imagine unicorn. The concept of unicorn is real because there exists an arrangement of matter that we can define as unicorn.

Almost You are getting close.
Take "honor". There is no arrangement of matter that let's you define precisely what honor is, and yet, (people claim) it is real. The key is the ability to communicate what honor is, i.e. you can define it with words, signs, logic, whatever.

This concept will come back below.

As for the "rearrange matter" definition, that's horrible when dealing with math. How do you represent the infinite dimensional vector spaces? You only have 3 dimensions to work with. What about the Banach-Taski theorem?

Because it's defined so. I'm suggesting that in physical reality there is no meaning that e.g.:$\varepsilon =infinitesimal,\frac{\varepsilon^{2} }{\varepsilon}=\varepsilon$

Real numbers don't have physical meaning either. I'll get to this in more depth below.

What definition do you mean? You mean the limit derivation definition? What exercises do you mean?

This:
Question: Find the derivative at x=1 of the function $f(x) = x^2$ using first principles.
Answer: Let $h$ be a real number satisfying $h\neq0$. Then
f(1+h)-f(1) = (1+h)^2-1^2 = 1 + 2h + h^2 - 1 = 2h+h^2
\frac{f(1+h)-f(1)}{h} = \frac{2h+h^2}{h} = 2+h
Then taking the limit
\lim_{h\to0} \frac{f(1+h)-f(1)}{h} = \lim_{h\to0}2+h = 2.
QED

Notice there are no infinitesimals anywhere.

I thought that we use reals because real numbers have meaning in physical reality.

Hardly. As others have said, we use it because it is convenient.

But the real numbers consist out of way more numbers than the natural numbers. In fact, most real numbers can't even be defined or described.

This is what I want to get to.
π is an irrational number. This means that it's decimal (or binary or any radix you want) representation cannot be written using finite symbols. But that's not a problem, because we can calculate the decimal representation to any degree of precision we want. In fact with hexadecimal representation the BBP formula allows us to calculate the n^th digit without calculating the first n-1 digits.

Real numbers that have an algorithm to calculate the their representation are called computable real numbers. And now we have Chaitin's constant. It's real but doesn't have such an algorithm. No problem we can still define this constant using a finite string of logical symbols. This set of real numbers are the definable real numbers.

Here's the kicker: the definable reals are countable. I'm assuming you know that reals are uncountable, hence the majority of real number cannot be defined. We know they exist, somehow, but no formula, equation, sentence, paragraph, arrangement of matter, or whatever will allow us to say "this is what this number is".

And that's why physical reality is bad justification for the real numbers.

 

[sludger13]

Take "honor". There is no arrangement of matter that let's you define precisely what honor is, and yet, (people claim) it is real. The key is the ability to communicate what honor is, i.e. you can define it with words, signs, logic, whatever.

Obviously there is an arrangement of matter that represents an honor. Strictly, there has to be some brain structure representing the meaning of honor. Human's brain is probably encoded by its evolution that it recognizes some specific arrangement of matter in timespace (also that is some specific people behavior) so we consider that behavior as honorable. Brain can also store those memories (= another arrangement of brain matter) so I can recall latter, what honor is. And finally, the brain structure can also be arranged so I can slightly remember the abstract meaning of honor without any specific conscious (situation, memory) of honor.
Also an honor, like everything else, can be precisely defined by some brain structure. However I do not claim people can make that definition at this time.

As for the "rearrange matter" definition, that's horrible when dealing with math. How do you represent the infinite dimensional vector spaces? You only have 3 dimensions to work with.

There has to be brain structure representing some set of finite-dimensional vectors (i.e. when you imagine some set of numbers), never mind you can 'compare' just $\mathbb{R}^{3}$ with the universe around us.
As for infinite-dimensional vectors, man can not obviously do that. In my opinion, the point is whether one feels that IDVS is sufficiently defined. Then it's clear what am I thinking of, although it cannot be directly imagined (unlike finite-dimensional vectors). Man can define IDVS using his (intuition, logical thinking...). By that definition, the concept of IDVS is uniquely specified. Also that definition represents the abstract meaning of IDVS. When I recall that definition = another brain structure... So far I couldn't think of better answer.

Question: Find the derivative at x=1 of the function $f(x) = x^2$ using first principles.
Answer: Let $h$ be a real number satisfying $h\neq0$. Then
f(1+h)-f(1) = (1+h)^2-1^2 = 1 + 2h + h^2 - 1 = 2h+h^2
\frac{f(1+h)-f(1)}{h} = \frac{2h+h^2}{h} = 2+h
Then taking the limit
\lim_{h\to0} \frac{f(1+h)-f(1)}{h} = \lim_{h\to0}2+h = 2.

Assumption:
with differentiable function, $0$ has always meaning of (what I call) lower infinity (it represents lower inf. in higher inf.). It never has the meaning of nothing.
Then this is for me the proof, that I can express the derivatives in higher infinities. That corresponds to the real slope of tangent in graph. Also just one real number axis describes the graph. In differential equations, I consider all derivatives in their highest infinity, that infinity I consider as real.

...And that's why physical reality is bad justification for the real numbers.

Nothing against it. Also part of real numbers :)

There is an abstract meaning (either created in my brain or even "deeper" existing in physical reality)...

I refer to the fact that elementary physical laws 'behaves mathematically'. Take Coulomb's law. The charged matter bahaves (= moves) according to that relation. Also does that relation exist in physical reality... or doesn't it?

 

[micromass]

I think it's time to lock this thread since it's getting too close to philosophy.