[guest4091]

#12

If we represent nothingness by zero then minus one is actually something.

Minus one represents a vector in the complex plane.

#14

Must there not be a difference between mathematical "nothing" and physical "nothing"?

Think of the symbol '0' as the empty container, i.e. containing nothing, and serving as a place holder for any number system. A minus integer now has no meaning, since there are no elements to remove. This assumes the symbol '-' denotes the operation of subtraction, whereas in #12 it denotes direction, a 2nd attribute of a vector. Integers do not have direction.
#20

If that is the case, it must be very loosely related, as infinity is not measurable

It began with counting, with the natural/whole numbers, for purposes of expressing multiplicity, family, livestock, commerce, etc., which is the simplest form of measurement. After someone demonstrated there couldn't be a largest integer, 'thinkers' speculated on determining a size for an 'infinite' set, as is done with finite sets. This is typical deduction, conceptualizing new things in terms of things currently understood *, a direct violation of the anonymous proverb, 'a brick is not made of smaller bricks'. If they continued adding more elements to the set, eventually (when?) it would become infinite. This introduces time as a factor. If a calculation algorithym requires an interval of time for each step, then ‘without limit’ translates to ‘always incomplete’,or impossible.
Consider Dedekinds’ method of forming an irrational number. If you pause at step n, you always have more cuts to make than the n you have done.. You are not making any progress to a final answer.

*A plumber could explain electric circuits in terms of fluid dynamics, but that doesn't mean electricity is water.

#21

The set of counting numbers is (countably) infinite.
ie you can count how many of them there are by drawing a 1 to 1 correspondence with the counting numbers (which is particularly easy to do, in this case).
There are an infinite number of numbers in this set.

To count is to quantify, typically for the purpose of finding a value that represents the size or extent of something.
Finite and measurable are synonyms for quantifiable. So we have another contradiction in terms. From the definition of 'infinite', infinite number equates to immeasurable number. Is it really a '1 to 1' correspondence?
My conclusion is, the mind lacks the ability to conceptualize the idea of something with a beginning and no end.
Assignment for today:
How do you measure a stick with only one end?

[jeffreyH]

The stick is not infinite in length since it terminates at one end. The is no such thing as half of infinity so there is a dilemma here. This is always the problems with human conceptions of infinity.

[evan_au]

From the definition of 'infinite', infinite number equates to immeasurable number. Is it really a '1 to 1' correspondence?
My conclusion is, the mind lacks the ability to conceptualize the idea of something with a beginning and no end.
Assignment for today:
How do you measure a stick with only one end?

Ok, I'll bite....

Lets take the integers (positive and negative whole numbers): 0, ±1, ±2, ±3, ±4..... etc
This could be considered a "ruler with two ends".

How do you put this in 1-1 correspondence with the non-negative integers: 0, 1, 2, 3, 4....etc? (which you could consider to be a "ruler with one end")

There are many ways to do it, but a simple one is:
- 0 (in non-negative integers) corresponds to 0 (in integers)
- even number n (in non-negative integers)  corresponds to n/2 (in integers), eg 54 corresponds with 27
- odd number n+1 (in non-negative integers)  corresponds to -n/2 (in integers), eg 55 corresponds with -27

By using a pronumeral "n", or the abbreviation "etc", I can make an infinite number of true statements in a single statement, and so I can prove 1-1 correspondence for an infinite number of cases - without consuming all the electrons in the universe (and still failing).

[Bill S (OP)]

In my view, change is constant, and the product of the forces that govern the interaction between matter and energy in said space.

OK with that; but you still have divisions of infinity.  If you remove one such division, what are you left with?  Infinity -1?

[Bill S (OP)]

The stick is not infinite in length since it terminates at one end. The is no such thing as half of infinity so there is a dilemma here. This is always the problems with human conceptions of infinity.

It will be fascinating to see what responses you get to this.  I have been saying this in various guises for several years, in this and other forums.  Someone always comes up with a mathematical “answer” that doesn’t really address the issue.

[Bogie_smiles]

OK with that; but you still have divisions of infinity.  If you remove one such division, what are you left with?  Infinity -1?

I didn’t include “divisions” of infinity in my definition. I think that if you have an infinite universe, there are no divisions of it, that I know of. Give me an example.

[Bill S (OP)]

I can make an infinite number of true statements......

Let's make a bid for clarity:  Is infinity a number?   What is "an infinite number of" anything?

[Bill S (OP)]

  I think that if you have an infinite Universe, there are no divisions of it, that I know of.

Agreed.  However, if you admit change, then you have the universe before, and after, each change; thus you have divisions. I would argue that in infinity there can be no change.

If the cosmos is infinite, there can be no change within the cosmos.  You may ask how I equate that with the idea that the Universe is "part" of the cosmos, and the Universe, manifestly, changes.  Hopefully we'll get there. but res unum post alium

[Bogie_smiles]

Agreed.  However, if you admit change, then you have the universe before, and after, each change; thus you have divisions. I would argue that in infinity there can be no change.

If the cosmos is infinite, there can be no change within the cosmos.  You may ask how I equate that with the idea that the Universe is "part" of the cosmos, and the Universe, manifestly, changes.  Hopefully we'll get there. but res unum post alium

My definition of infinity, in regard to the universe, does have language that can avoid such an inconvenient interpretation. It is in the last phrase, when talking about the invariant natural laws … “that also restrain the possibilities of what is and what can be.” According to my interpretation, your interpretation that, "there can be no change" within the cosmos, could be a violation of the set of invariant natural laws.

For example, since it is in accord with the natural laws that when there is the presence of matter and energy, there is going to be continual change, then it is natural that there would be change within the “one boundless, eternal, contiguous space”.

[Bill S (OP)]

According to my interpretation, your interpretation that, "there can be no change" within the cosmos, could be a violation of the set of invariant natural laws.

It could also be a violation of some unknown divine edict, but unless we know what that invariant natural law, or divine edict might be, how can we base a scientific argument on either?

For example, since it is in accord with the natural laws that when there is the presence of matter and energy, there is going to be continual change, then it is natural that there would be change within the “one boundless, eternal, contiguous space”.

Let’s consider a single change in “the one boundless, eternal, contiguous space”.  This space has always existed in a specific state.  Then comes change; after which its state is different. 

Is this not like Jeffrey’s stick; it “is not infinite in length since it terminates at one end”?

[Bogie_smiles]

It could also be a violation of some unknown divine edict, but unless we know what that invariant natural law, or divine edict might be, how can we base a scientific argument on either?

A divine edict is not science because the supernatural is not recognized by the scientific method. Anything that seems Supernatural, has natural causes that we don’t yet understand.

Let’s consider a single change in “the one boundless, eternal, contiguous space”.  This space has always existed in a specific state.  Then comes change; after which its state is different.

Where does the idea that the contiguous space has always existed “in a specific state”, come from? It has always existed and has always encompassed all there is, all matter, energy, everything, in one boundless, eternal, contiguous space, and the straight forward interpretation is that such a universe would be in more than one state, and would probably encompass all of the possible states permitted by the invariant natural laws.

Is this not like Jeffrey’s stick; it “is not infinite in length since it terminates at one end”?

Infinite and eternal, no beginning, no end, (and nothing Supernatural), is my interpretation of the stated definition of a working concept of Infinity/eternity.

There must be better definitions out there; someone will post one if they want to improve on mine.


Edit: Let me know if this line of discussion goes beyond the intended scope of this sub-forum.

[Bill S (OP)]

Bogie_smiles, I'm not ignoring your post, but am finding breathing (and thinking ) a bit difficult today, and want to be able to do justice to your points.

I might look for some "reprise" points as an easy option, and to try to stop them from slipping into oblivion.

[Bill S (OP)]

“It’s reprise time folks”    Yes, that was Hughie Green, but who’s old enough to remember that?

We won't ever know if there was ever nothing. We can assume that there wasn't ever nothing but that is by choice.

Has anyone demonstrated a mechanism by which something can emerge from nothing?

I know books have been written about it, but doesn't the "nothing" always turn out to be "something"?

[evan_au]

I can make an infinite number of true statements......

I do not claim that I can make an infinite number of statements that are particularly novel or profound.
- But it is useful in mathematics, where you can concisely make statements that are true for all numbers.
- eg a+a = 2*a is true for all real numbers.

Let's make a bid for clarity:  Is infinity a number?   What is "an infinite number of" anything?

I would say that "infinity is not a specific number; it is larger than any specific number.".

As Cantor showed, there are different kinds of infinity:
- There are an infinite number of integers. But you can count them. This is called ℵ₀
- Some kinds of infinity (eg the number of "real" numbers) is so much larger than the number of integers that you cannot count them. This is called ℵ₁
- There are even larger infinities.
See: https://en.wikipedia.org/wiki/Aleph_number

I would also say that we don't have any examples of an infinite number of real, physical things. So the physical world does not map directly onto mathematics.

[yor_on]

Eh. Evan :)

If we had, wouldn't that change it into something finite?

[jeffreyH]

For the infinite series of positive integers can we start counting from the highest and count downwards? Would we ever reach zero? Could we find a highest number to start from? If so isn't that a finite series?

[Bogie_smiles]

For the infinite series of positive integers can we start counting from the highest and count downwards?

There is no highest, it is an infinite series to the high side.

Would we ever reach zero?

If you could find a starting point to count backwards from, you could reach zero, but the premise of staring from the highest number in an infinite series is flawed.

Could we find a highest number to start from? If so isn't that a finite series?

That is a no and a yes, .

[jeffreyH]

For the infinite series of positive integers can we start counting from the highest and count downwards?

There is no highest, it is an infinite series to the high side.

Would we ever reach zero?

If you could find a starting point to count backwards from, you could reach zero, but the premise of staring from the highest number in an infinite series is flawed.

Could we find a highest number to start from? If so isn't that a finite series?

That is a no and a yes, .

Exactly! So how is an infinite series countable if you can't count from the highest to lowest?

[Bogie_smiles]

For the infinite series of positive integers can we start counting from the highest and count downwards?

There is no highest, it is an infinite series to the high side.

Would we ever reach zero?

If you could find a starting point to count backwards from, you could reach zero, but the premise of staring from the highest number in an infinite series is flawed.

Could we find a highest number to start from? If so isn't that a finite series?

That is a no and a yes, .

Exactly! So how is an infinite series countable if you can't count from the highest to lowest?

Its not.


Edit to my working concept of infinity/eternity: That definition could be expanded to include an infinite mathematical series, as well … A mathematically infinite series is simply an ever increasing or ever decreasing series, or both. In the case of both, counting forward or backward from any selected starting point will never result in reaching the beginning or the end of the series.


Even if you count by tens, lol.

[evan_au]

For the infinite series of positive integers can we start counting from the highest and count downwards?

Yes; the difference in the mathematical equation is as follows:
= 2  (Oops!   2 is the answer if you start from n=0, not n=1 )

= 2

You basically reverse the upper & lower limits.
But the answer remains the same (2).

Would we ever reach zero?

Only by doing an infinite number of operations at once.
Mathematics has a number of ways of doing this, including the ∑ operator.

Could we find a highest number to start from? If so isn't that a finite series?

Yes (by approximation) and Yes.
For example, the standard mathematical infinite series for arctan converges fairly slowly, especially around π/4 radians.
But most people don't want their calculations done to infinite precision - most people are happy with 8-18 digits, most of the time (unless they are actually trying to break the record for generating the most digits for π).

There are various mathematical transformations that allow you to approximate an infinite series to a specific number of decimal places, using a finite series instead of an infinite series. That gives you a predictable execution time for your mathematical library, so your self-driving car doesn't suddenly become unresponsive when the road swings around to bearing π/4.
See, for example: https://en.wikibooks.org/wiki/Trigonometry/For_Enthusiasts/Chebyshev_Polynomials

But approximation is more in the realm of applied mathematics, rather than pure mathematics.
The real world is so complicated (and not so well understood) that sometimes approximations are all we have.