(January 18, 2022 at 8:12 pm)polymath257 Wrote:

(January 18, 2022 at 5:45 pm)brewer Wrote: If zero is not 'natural', where does than leave negative numbers?

Yeah Poly, I'm looking to you to hurt my math brain again.

The usual progression is something like this:

1. Natural numbers: 0,1,2,3,4,....

2. Integers: Either a natural number or the negative of a natural number, so ..,-3, -2, -1, 0, 1, 2, 3,...

3. Rational numbers: Fractions, one integer divided by a non-zero integer. 2/3, -5/8, 13/2, 4=4/1, -123/43,....

4. Real numbers: intuitively, numbers with (possibly infinite) decimal expansions. pi, e, sqrt(2), ....

5. Complex numbers. If i is the square root of -1, then a complex number is one of the form a+bi where a and b are real numbers.

These are the basic number systems, but there are a host of others.

Algebraic numbers: those that are roots of some polynomial with integer coefficients. sqrt(2) is algebraic. so is sqrt(2)+cbrt(5).

Gaussian integers: those complex numbers a+bi where a and b are integers. these have many properties in common with the integers.

These are the usual algebraic number systems.

But we can go in a different direction. All natural numbers are also 'cardinal numbers': these count the 'number of things' in sets. The natural numbers are, specifically, the
*finite* cardinal numbers.

But there are infinite cardinal numbers as well. These are studied in set theory and usually not as algebraic structures because they lack many of the 'nice' properties of the other systems above. if you are trying to say infinity is a number, this is what you are probably talking about.

Ordinal numbers: instead of the 'size' of a set, we put an 'order structure' on the set and compare different order structures. Usually, we do this with order structures that are 'nice' in the sense of being 'well-ordered' (a technical term that I can go into if anyone wants). The natural numbers are, again, the finite ordinal numbers. But ordinal numbers have a much more complex structure than cardinal numbers do. They also fail to obey certain fairly natural algebraic properties, so are usually studied by set theorists as well.

In standard set theory, following Von Neumann, every cardinal number is an ordinal number, but not vice versa.

Ouch, ouch, ouch, my late high school-early undergrad brain hurts. I expect a delivery of fentanyl and midazolam. Not propofol, no Jackson for me.

Just for fun, does negative or positive ever apply to zero?

I wonder...the proposition "Out of nothing, nothing comes," seems self-evident and serves as a foundation level defeater in ontology. That would seem to have some bearing on whether it is real or natural. And what's natural about numbers? According to some, numbers, like all mathematical objects, are artifices that sometimes inexplicably align consistently with observed phenomena. Such as those would by definition not be natural, but artificial.

(January 18, 2022 at 9:17 pm)brewer Wrote:

(January 18, 2022 at 8:12 pm)polymath257 Wrote: The usual progression is something like this:

1. Natural numbers: 0,1,2,3,4,....

2. Integers: Either a natural number or the negative of a natural number, so ..,-3, -2, -1, 0, 1, 2, 3,...

3. Rational numbers: Fractions, one integer divided by a non-zero integer. 2/3, -5/8, 13/2, 4=4/1, -123/43,....

4. Real numbers: intuitively, numbers with (possibly infinite) decimal expansions. pi, e, sqrt(2), ....

5. Complex numbers. If i is the square root of -1, then a complex number is one of the form a+bi where a and b are real numbers.

These are the basic number systems, but there are a host of others.

Algebraic numbers: those that are roots of some polynomial with integer coefficients. sqrt(2) is algebraic. so is sqrt(2)+cbrt(5).

Gaussian integers: those complex numbers a+bi where a and b are integers. these have many properties in common with the integers.

These are the usual algebraic number systems.

But we can go in a different direction. All natural numbers are also 'cardinal numbers': these count the 'number of things' in sets. The natural numbers are, specifically, the
*finite* cardinal numbers.

But there are infinite cardinal numbers as well. These are studied in set theory and usually not as algebraic structures because they lack many of the 'nice' properties of the other systems above. if you are trying to say infinity is a number, this is what you are probably talking about.

Ordinal numbers: instead of the 'size' of a set, we put an 'order structure' on the set and compare different order structures. Usually, we do this with order structures that are 'nice' in the sense of being 'well-ordered' (a technical term that I can go into if anyone wants). The natural numbers are, again, the finite ordinal numbers. But ordinal numbers have a much more complex structure than cardinal numbers do. They also fail to obey certain fairly natural algebraic properties, so are usually studied by set theorists as well.

In standard set theory, following Von Neumann, every cardinal number is an ordinal number, but not vice versa.

Ouch, ouch, ouch, my late high school-early undergrad brain hurts. I expect a delivery of fentanyl and midazolam. Not propofol, no Jackson for me.

Just for fun, does negative or positive ever apply to zero?

Ooh, ooh, I know this one! NO.

(January 18, 2022 at 9:34 pm)Neo-Scholastic Wrote: I wonder...the proposition "Out of nothing, nothing comes," seems self-evident and serves as a foundation level defeater in ontology. That would seem to have some bearing on whether it is real or natural. And what's natural about numbers? According to some, numbers, like all mathematical objects, are artifices that sometimes inexplicably align consistently with observed phenomena. Such as those would by definition not be natural, but artificial.

What's the alternative? That the laws of nature not be describable by some mathematical models? The term for such a world would be "magic".

As for "nothing" no one has, to my satisfaction, ever provided any physical description of that. To use "nothing" in a sentence is to give it meaning, if only as an abstraction. One might as well ask what color Saturday is.

(January 18, 2022 at 9:17 pm)brewer Wrote: Ouch, ouch, ouch, my late high school-early undergrad brain hurts. I expect a delivery of fentanyl and midazolam. Not propofol, no Jackson for me.

Just for fun, does negative or positive ever apply to zero?

Depends a bit on the book used.  One way of resolving ambiguities is as follows:

Positive: >0
Negative: <0
Non-negative: >=0
Non-positive: <=0

So, 0 is not negative, not positive, but is non-negative and non-positive.

(January 18, 2022 at 9:55 pm)Jehanne Wrote:

(January 18, 2022 at 9:34 pm)Neo-Scholastic Wrote: I wonder...the proposition "Out of nothing, nothing comes," seems self-evident and serves as a foundation level defeater in ontology. That would seem to have some bearing on whether it is real or natural. And what's natural about numbers? According to some, numbers, like all mathematical objects, are artifices that sometimes inexplicably align consistently with observed phenomena. Such as those would by definition not be natural, but artificial.

What's the alternative? That the laws of nature not be describable by some mathematical models? The term for such a world would be "magic".

As for "nothing" no one has, to my satisfaction, ever provided any physical description of that. To use "nothing" in a sentence is to give it meaning, if only as an abstraction. One might as well ask what color Saturday is.

Well...it seems to me that nothingness literally means an absence of physicality.

Quote:Well...it seems to me that nothingness literally means an absence of physicality.

Nope 

(January 18, 2022 at 9:17 pm)brewer Wrote:

(January 18, 2022 at 8:12 pm)polymath257 Wrote: The usual progression is something like this:

1. Natural numbers: 0,1,2,3,4,....

2. Integers: Either a natural number or the negative of a natural number, so ..,-3, -2, -1, 0, 1, 2, 3,...

3. Rational numbers: Fractions, one integer divided by a non-zero integer. 2/3, -5/8, 13/2, 4=4/1, -123/43,....

4. Real numbers: intuitively, numbers with (possibly infinite) decimal expansions. pi, e, sqrt(2), ....

5. Complex numbers. If i is the square root of -1, then a complex number is one of the form a+bi where a and b are real numbers.

These are the basic number systems, but there are a host of others.

Algebraic numbers: those that are roots of some polynomial with integer coefficients. sqrt(2) is algebraic. so is sqrt(2)+cbrt(5).

Gaussian integers: those complex numbers a+bi where a and b are integers. these have many properties in common with the integers.

These are the usual algebraic number systems.

But we can go in a different direction. All natural numbers are also 'cardinal numbers': these count the 'number of things' in sets. The natural numbers are, specifically, the
*finite* cardinal numbers.

But there are infinite cardinal numbers as well. These are studied in set theory and usually not as algebraic structures because they lack many of the 'nice' properties of the other systems above. if you are trying to say infinity is a number, this is what you are probably talking about.

Ordinal numbers: instead of the 'size' of a set, we put an 'order structure' on the set and compare different order structures. Usually, we do this with order structures that are 'nice' in the sense of being 'well-ordered' (a technical term that I can go into if anyone wants). The natural numbers are, again, the finite ordinal numbers. But ordinal numbers have a much more complex structure than cardinal numbers do. They also fail to obey certain fairly natural algebraic properties, so are usually studied by set theorists as well.

In standard set theory, following Von Neumann, every cardinal number is an ordinal number, but not vice versa.

Ouch, ouch, ouch, my late high school-early undergrad brain hurts. I expect a delivery of fentanyl and midazolam. Not propofol, no Jackson for me.

Just for fun, does negative or positive ever apply to zero?

In computer science, yes.

(January 18, 2022 at 10:56 pm)Neo-Scholastic Wrote:

(January 18, 2022 at 9:55 pm)Jehanne Wrote: What's the alternative? That the laws of nature not be describable by some mathematical models? The term for such a world would be "magic".

As for "nothing" no one has, to my satisfaction, ever provided any physical description of that. To use "nothing" in a sentence is to give it meaning, if only as an abstraction. One might as well ask what color Saturday is.

Well...it seems to me that nothingness literally means an absence of physicality.

You're defining it in terms of what it is not as opposed to what it is, even if such is only as a concept.

Really, we're going to have a debate over nothing? No thanks.