1. ## Discreet versus continuum

(sorry, don't know in which section this post goes)

I just took a look at this: Continuity and Infinitesimals (Stanford Encyclopedia of Philosophy) and this https://en.wikipedia.org/wiki/Infinitesimal.

A lot of it I can't understand, especially the math (meaning what is not explained with language concepts, but math notation).

I have one observation about the mathematical continuum idea though. Someone recently asked me: "Do you understand that the line segment, say, (0, 1) is a continuum like time, and not made of discrete blocks or atoms/molecules like physical space? "

I think of both time and space as separate things, and examples that each function like a continuum, but which we can also measure discretely, that is, with discrete units. Whether you look at a line as one "long" thing or as a sequence of discreet, adjoining elements depends on your approach.

From the little I understood of the concept of infinitesimal and continuum, infinitesimals no longer function in the same way as "normal" discrete numbers, like integers on a natural number line.

And to me, that paints a curious picture. Because, if you take the abstract principle of infinite division of something, you should be able to divide it infinitely, and the concept of the elements you are working with shouldn't change. It's always the same, just smaller. But apparently this can't be done with numbers?

From what I understood (correct me if I'm wrong), you take a number, which is discreet, and you keep dividing it. Now this number at some point is not going to have the same properties as the number you started with? It's not going to function the same way, it's not going to be discreet (or as discreet)? So it's not just a smaller version of the bigger number, but it changed into another being.

The funny thing is, this is what happens when we are dealing with matter. As you get into the smaller, you find small elements that are different than the bigger whole (like atoms).

So this change in the element itself ruptures the idea of infinite division in an abstract way, and of infinity itself, in the sense that the process can't go on in the same way forever.

But if math is abstract, shouldn't it be the realm where one could divide a number infinitely and the number you get is simply a smaller, but exact (having the same properties), replica of the number you just divided?

Apparently you can do this with an abstract thing or distance, but not with numbers.

Like the equivalent of the distance in Zeno's taking a step paradox. You start with a distance. You divide it in half, it's still a distance, that is, it's greater than zero. And I could measure it discreetly, with discreet units. Always. It would never be non-discreet.

Now maybe I didn't understand what this infinitesimal / continuum was about, but that's what I thought I understood.

2. ## Re: Discreet versus continuum

Originally Posted by Alessandra
I just took a look at this: Continuity and Infinitesimals (Stanford Encyclopedia of Philosophy) and this https://en.wikipedia.org/wiki/Infinitesimal.
A lot of it I can't understand, especially the math (meaning what is not explained with language concepts, but math notation).
I have one observation about the mathematical continuum idea though. Someone recently asked me: "Do you understand that the line segment, say, (0, 1) is a continuum like time, and not made of discrete blocks or atoms/molecules like physical space? "
I think of both time and space as separate things, and examples that each function like a continuum, but which we can also measure discretely, that is, with discrete units. Whether you look at a line as one "long" thing or as a sequence of discreet, adjoining elements depends on your approach.
Please do not confuse mathematics with physics. Mathematics is not a description of reality.
It is well to understand this:
“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” Albert Einstein in Geometry & Experience, 1929.

3. ## Re: Discreet versus continuum

Originally Posted by Alessandra
From what I understood (correct me if I'm wrong), you take a number, which is discreet, and you keep dividing it. Now this number at some point is not going to have the same properties as the number you started with? It's not going to function the same way, it's not going to be discreet (or as discreet)? So it's not just a smaller version of the bigger number, but it changed into another being.
There's a lot that is wrong with this. Reals do not turn into infinitesimals simply by dividing them. And infinitesimals have no analogue in reality (the irrational numbers almost certainly do not either). They are purely a mathematical tool for understanding certain areas of mathematics in the real numbers.

4. ## Re: Discreet versus continuum

And infinitesimals have no analogue in reality (the irrational numbers almost certainly do not either)
Surely the square root of two comes up in physical things, things where there is a right angled triangle and a hypotenuse of length sqrt(2). Also pi could be found in reality by looking at circular things like gravitational fields.

I would take it a step further and say that indescribable numbers have no physical basis, maybe also non-computable numbers too

5. ## Re: Discreet versus continuum

There are no reason to suppose that there are any perfect circles or perfect squares in the real world. There are some good reasons to suspect that space may not even be continuous.

We will never know if any irrational number really did occur, because all our measurements are approximations (and always will be) which makes them rational.

Of course, if space turns out to be continuous after all, then the likelihood is that there are no rational distances, but it is equally unlikely that there will be any examples of $\pi$ or $\sqrt2$ either.

6. ## Re: Discreet versus continuum

Originally Posted by Archie
... but it is equally unlikely that there will be any examples of $\pi$ or $\sqrt2$ either.
How dare you insult $\displaystyle \pi$ on this very very special day!

-Dan

7. ## Re: Discreet versus continuum

There's nowt wrong with $\pi$. It's the universe that is needlessly complicated and subject to experimental and measurement errors.

Not that I'm very interested in $\pi$ day. Not until the date/time actually has an infinite number of digits, anyway.