Are Infinitesimals Really Numbers?

[Write4U]

Yes that's true. There are real numbers for which we can not have any symbols or descriptions or ways of specifying them.

This is a bit off-topic from infinitesimals. There are the non-computable numbers. These are real numbers that can not be described by any algorithm or finite-length string of symbols. There are many noncomputable real numbers. If you pick a random real, it will almost certainly be noncomputable. It's a little mysterious.

By the way "measurable" is a technical term in math that means something else, so best to avoid it.




If I'm understanding you, you're making the distinction between a number and any of its many possible representations. For example 2 + 2, 4, and 10 (base 4) are different expressions that both point to the same number. The number itself is an abstraction.Much food for thought.



Yes in this case that's true. But most real numbers don't have closed-form representations. All the familiar numbers we use do have closed-form representations though. For example the digits of pi or sqrt(2) can be cranked out by a computer program, and a program is a finite string of symbols. We can think of a program as a closed form for the number whose digits it cranks out. But since there are fewer programs than real numbers, there are real numbers without closed forms.

To be fair, nothing in this thread is about the physical world. We're talking about the mathematical abstraction of the real numbers. As far as we know, real numbers can't be instantiated in the physical world because it takes an infinite amount of information to specify most real numbers.

It's always important to distinguish math from physics.

Thank you for that comprehensive response. Much food for thought.

 

[TheFrogger]

Ahhh truthseeker. It would never arrive.

 

[Write4U]

Ahhh truthseeker. It would never arrive.

Is a zero mass muon, infinitely small? Does it generate a wave function?
Can an infinitely small particle acquire mass from it's speed alone?
In a *quantum* world , can there be such a thing as infinitely small?

 

[someguy1]

Is a zero mass muon, infinitely small? Does it generate a wave function?
Can an infinitely small particle acquire mass from it's speed alone?
In a *quantum* world , can there be such a thing as infinitely small?

Those are questions of physics, not math. I don't know what physicists say about point particles these days. Aren't they wiggles in some probability space or something? I don't know much physics.

But I doubt anything could be "infinitely" small in the real world. We can't even sensibly talk about anything below the Planck length so how could we know? Maybe some physics people can chime in here. What do physicists think about really tiny particles?

 

[TheFrogger]

I have a similar thought. Should we walk TO a location, only going half the distance each time, would we ever arrive?

The formula is something like:

a=0
b=1
1:
b=b/2
a=a+b
goto 1

or...

1-(2^x)

We would never arrive. The formula is infinite, despite the fact computers round the number to 1 at about 35/36 tries.

 

[DaveC426913]

I have a similar thought. Should we walk TO a location, only going half the distance each time, would we ever arrive?

We would. This is Zeno's Paradox. And it's been resolved.

 

[someguy1]

We would. This is Zeno's Paradox. And it's been resolved.

I'm not one who agrees with that. Of course the standard answer is that calculus shows that $$\sum_{i=1}^n \frac{1}{2^n} = 1$$ end of story move along nothing to see here.

I am not satisfied. That is the mathematical answer, of course. But Zeno was talking about the physical world. As of the present moment physics has no referent for summing up an infinite series or taking a limit. We're told it's not sensible to even ask what happens below the Planck scale. Whether the world is discrete or continuous, and whether the modern formalism of the real numbers is the right model of the world, are open questions. These matters are not settled. The mathematical theory of convergent infinite series does not resolve Zeno's paradoxes.

That is my opinion, but if someone can talk me out of it I'd be grateful. It's tedious to be the one to claim that calculus doesn't resolve Zeno when so many people think it does.

 

[DaveC426913]

I'm not one who agrees with that. Of course the standard answer is that calculus shows that $$\sum_{i=1}^n \frac{1}{2^n} = 1$$ end of story move along nothing to see here.

I am not satisfied. That is the mathematical answer, of course. But Zeno was talking about the physical world.

Since we know it works in the physical world, and the calculus simply describes it, what partly exactly do you find hard to believe?

It works in practice; it works in theory. What's left?

 

[Write4U]

I have a similar thought. Should we walk TO a location, only going half the distance each time, would we ever arrive?

When we walk we do not reduced the distance between points, we relatively increase the distance covered by our step which (at that scale) keeps getting larger until is larger than half the entire remaining distance between points......

 

[someguy1]

Since we know it works in the physical world, and the calculus simply describes it, what partly exactly do you find hard to believe?

It works in practice; it works in theory. What's left?

I'm afraid I don't see how it works in theory. No physical theory justifies the notion that there are infinitely many "points," whatever they are, between two locations in space. To be clear:

* No physical theory discusses or posits an infinite quantity of quarks, atoms, strings, loops, or subatomic gizmos of any kind; and

* No physical theory defines something called a point.

These are purely mathematical notions that have no known referents in the physical world according to the best contemporary physical theories.

And you can't add up those really tiny distances, the ones smaller than the Planck length, because in physics we can't speak meaningfully about them at all.

Once you separate the math from the physics. you have no infinite sums. In my minority opinion of course. Most people agree with you.

 

[Write4U]

I'm afraid I don't see how it works in theory. No physical theory justifies the notion that there are infinitely many "points," whatever they are, between two locations in space. To be clear:

I agree. if QM is correct and information propagates in quanta, there must be space between the change of states.
In a binary universe (on/off) reality would be manifest only 1/2 the time, the other 1/2 would be in *quantum suspension* (implicate potential) and not manifest.
All of this would be beyond our experience.

 

[DaveC426913]

These are purely mathematical notions that have no known referents in the physical world according to the best contemporary physical theories.

The math shows it working in theory; the physical action shows it working physically.
I ask again: what is left?

 

[Write4U]

The math shows it working in theory; the physical action shows it working physically.
I ask again: what is left?

Question: Instead of asking for a *number*, might it be more productive to ask if infinitesimals have *values*. No value = 0

 

[someguy1]

The math shows it working in theory; the physical action shows it working physically.
I ask again: what is left?

From https://en.wikipedia.org/wiki/Planck_length :

In some forms of quantum gravity, the Planck length is the length scale at which the structure of spacetime becomes dominated by quantum effects, and it is impossible to determine the difference between two locations less than one Planck length apart.

(My emphasis on that last phrase).

Now the Planck length is given as $$1.616229(38) \times 10^{-35}$$ meters. How many Zeno steps is this?

For a back-of-the-envelope Sunday morning calculation we can just call the Planck length $$10^{-36}$$. Then

$$log_2 (10^{-36}) = -36 \times log_2(10) \approx -36 \times 3.3219 \approx -119.56 \approx -120$$.

So if we are trying to cover a distance of one meter in discrete steps, each step covering exactly half the remaining distance, it will take us a mere 120 steps or so before the distances involved are so small that it is meaningless to reason about them. That's the point where known physics diverges from the mathematical theory of the real numbers.

 

[DaveC426913]

From https://en.wikipedia.org/wiki/Planck_length :



(My emphasis on that last phrase).

Now the Planck length is given as $$1.616229(38) \times 10^{-35}$$ meters. How many Zeno steps is this?

For a back-of-the-envelope Sunday morning calculation we can just call the Planck length $$10^{-36}$$. Then

$$log_2 (10^{-36}) = -36 \times log_2(10) \approx -36 \times 3.3219 \approx -119.56 \approx -120$$.

So if we are trying to cover a distance of one meter in discrete steps, each step covering exactly half the remaining distance, it will take us a mere 120 steps or so before the distances involved are so small that it is meaningless to reason about them. That's the point where known physics diverges from the mathematical theory of the real numbers.

OK, so you can't move a half a Planck length.

Your attempt to reduce each step by half fails at that scale; the smallest distance you can move is one Planck length.

All you've done is conclude that the physical world does not go smaller than Planck units - something we already know.

 

[someguy1]

OK, so you can't move a half a Planck length.

A stronger statement is at work: The question of "moving half a Planck length" is not even meaningful. It doesn't have a truth value at all. It's not true, it's not false. It's not a meaningful question about the world.

OK, so you can't move a half a Planck length.
Your attempt to reduce each step by half fails at that scale; the smallest distance you can move is one Planck length.

All you've done is conclude that the physical world does not go smaller than Planck units - something we already know.

I did no such thing. I don't see how you concluded that from what I wrote. I took what's known about physics, and I calculated the number of Zeno steps after which no more lengths can be added, because at that scale there are no meaningful lengths.

Math $$\neq$$ physics.

 

[Write4U]

Hmmmmm.....maybe math guides physics? IMO, this the uncertainty effect. The guiding maths becomes only measurable after the quantum event. But all physical functions and reactions (patterns) have a form of mathematical logic, if your perspective of abstract universal potentials that can create everything which is mathematically permitted can be imagined, the mathematical functions themselves are relatively simple.
In that context, even "adaption" is a mathematical (chemical) readjustment of cells and body functions for maximum efficiency in the new environment.

IMO, Intelligence is the ability of an organism to *cognize* (understand) and utilize information from their immediate environment.
Specialization (a branch of evolutionary functions) is relative to locality, Variety and adaptive powers (a different branch of evolutionary functions) is global in scope.