What Are Numbers?

[bhobba]

Introduction
When doing mathematics,  we usually take for granted what natural numbers, integers, and rationals are. They are pretty intuitive.   Going from rational numbers to reals is more complicated.   The easiest way at the start is probably infinite decimals.  Dedekind Cuts can be used to get a bit more fancy.  A Dedekind cut is a partition of the rational numbers into two sets, A and B, such that all elements of A are less than all elements of B, and A contains no largest rational number.  It corresponds to a point on the natural number line.  A is all the rationales to the left, and B is all the rationales to the right, including the point if it is rational.
But little niggles remain.
Cracks In Mathematics
Everyone has probably seen why x = .9999999….. = 1.  It’s simple.  10*.99999999999…. = 9.9999999…. 10*x – x = 9*x = 9 or x=1.   Everything sweet.   As an exercise, go to youtube, and you will see many videos on why it is true or not true.   The reason the debate rages is...

Continue reading...

 

[jbriggs444]

I see:
"We define 0 := ∅ and if $n = 0$ then we define n = n−1∪{n−1}."

Surely if $n \ne 0$ was intended.

 

[Mark44]

Operations On Integers.
The addition of two integers is defined as (a, b) + (c, d) = (a + c, b + d). Hence, for example, (1, 3) + (5, 2) = (6, 5). This will correspond to −2 + 3 = 1.

This is a weird definition, IMO. To compute your example (-2 + 3), that involves adding integers, you have to find pairs of numbers that add up to, respectively, -2 and 3. One possibility would be the sum of the ordered pairs (4, 6) + (8, 5). By your definition the result is (12, 11) = 1.

In short, in order to be able to add -2 and 3, you have to know that (4, 6) represents -2. That is, in order to calculate -2 + 3, you have to be able to calculate 4 - 6.

 

[pbuk]

I see:
"We define 0 := ∅ and if $n = 0$ then we define n = n−1∪{n−1}."

Surely if $n \ne 0$ was intended.

Actually I think $n \gt 0$ was intended. But this is still not correct because we haven't defined what $n - 1$ means. Instead this should read "We define $0 = \emptyset$ and we define $n + 1 = n \cup \{ n \}$", or IMHO preferably* "We define $0 = \emptyset$ and we define $S(n) = n \cup \{ n \}$".

This seems to be quoted from the article https://www.revistaminerva.pt/on-the-nature-of-natural-numbers/ which looks pretty weak to me on a number of points.

* I prefer introducing the notion of the successor because it avoids confusion between $n + 1$ as a successor and the addition $a + b; b = 1 = {0, {0}}$.

 

[pbuk]

I have some more comments:

• The "v" in von Neumann's surname should not be capitalized (unless it is the first letter of a sentence).
• You have used the "*" symbol for multiplication. It would be better to use $\LaTeX$ throughout, or failing that the Unicode multiplication symbol "×" or middle dot "·".
• Your second link on ZFC requires a login to medium.com which I happen to have but not everyone may want, and your first link requires a login to something called "brilliant.org" which I definitely do not want.
• You introduce WOLOG in the first quarter of the page but don't use it until the last quarter: this is confusing.
• I haven't seen subtraction introduced on the natural numbers before (rather than the first step in considering the integers), but if you are going to do this then I don't think defining $0 - 1 = 0$ is a very good start.
• I have not come across hyperrationals before (I am of course familiar with hyperreals). What are they useful for? Why do you choose to introduce them immediately after the rationals before completing the reals?
• Your first "WOLOG" paragraph under the heading "The Rationals Are Dense in the Reals" is incorrect because by stating $a < b$ you are losing generality because you are excluding $a = b$. Instead this clause should simply read "Let a and b be distinct real numbers $a < b$ (you don't need the "WOLOG" (I prefer w.l.o.g) prefix because $a$ and $b$ are already general labels so you are free to assign them so that $a < b$).
• You are also using capital letters here to denote both sets (M) and elements of sets (N), and you are also using small letters to denote elements (a, b). Again, use $\LaTeX$ or failing that Unicode characters ℕ, ℤ, ℚ, ℝ, ℂ or failing that use capitals only for sets. Also if not using $\LaTeX$ use HTML _subscript and ^superscript, either directly or via the WYSIWYG editor.
• Your second "WOLOG" paragraph under this heading jumps immediately to the conclusion that "there is a rational Q between R and R+ε, R < Q < R+ε". Why? You need to find one such rational (clue: $\exists n \in \mathbb N^+ : 1 / n < \varepsilon$).
• Again "WOLOG ε>0" is meaningless here. You should revise use of w.l.o.g: https://en.wikipedia.org/wiki/Without_loss_of_generality.
• Why do you only include Natural Numbers, Integers, and Rationals in the title when the article includes all of the Reals as well as other Hyperreals?
• I assume you are aware of Dedekind's essay Was sind und was sollen die Zahlen?, usually translated as What are numbers and what should they be?? I think it would be better if you referred (and compared the scope of your article) to Dedekind's - or chose a different title.
• There is already a good Insight on number systems: https://www.physicsforums.com/insights/counting-to-p-adic-calculus-all-number-systems-that-we-have/ and I find the path from natural numbers to reals in that article clear and rigorous. Did you consider starting your article from where @fresh_42's left off (he did not include the non-real hyperreals) rather than rewriting this material?

 

[symbolipoint]

A very different viewpoint on the titled topic, "What are Numbers" but which misses some of the other specified parts of the title: Numbers are adjectives to help give information about quantity.

 

[Algr]

A very different viewpoint on the titled topic, "What are Numbers" but which misses some of the other specified parts of the title: Numbers are adjectives to help give information about quantity.

Can the laws of mathematics be derived from the laws of grammar?

 

[symbolipoint]

Can the laws of mathematics be derived from the laws of grammar?

I really, really believe no. MAYBE a few people have something different to say.

 

[bhobba]

I see:
"We define 0 := ∅ and if $n = 0$ then we define n = n−1∪{n−1}."

Surely if $n \ne 0$ was intended.

Fixed.

 

[bhobba]

This is a weird definition, IMO. To compute your example (-2 + 3), that involves adding integers, you have to find pairs of numbers that add up to, respectively, -2 and 3. One possibility would be the sum of the ordered pairs (4, 6) + (8, 5). By your definition the result is (12, 11) = 1.

In short, in order to be able to add -2 and 3, you have to know that (4, 6) represents -2. That is, in order to calculate -2 + 3, you have to be able to calculate 4 - 6.

Yes, it is a little weird. The following may help:
https://amsi.org.au/teacher_modules/Integer.html

Thanks
Bill

 

[bhobba]

Actually I think $n \gt 0$ was intended. But this is still not correct because we haven't defined what $n - 1$ means. Instead this should read "We define $0 = \emptyset$ and we define $n + 1 = n \cup \{ n \}$", or IMHO preferably* "We define $0 = \emptyset$ and we define $S(n) = n \cup \{ n \}$".

This seems to be quoted from the article https://www.revistaminerva.pt/on-the-nature-of-natural-numbers/ which looks pretty weak to me on a number of points.

* I prefer introducing the notion of the successor because it avoids confusion between $n + 1$ as a successor and the addition $a + b; b = 1 = {0, {0}}$.

Fixed. Changed to 'We define 0 = ∅ and define n+1 = n∪{n}.'

 

[bhobba]

I have some more comments:

Wow - you raise a lot of interesting points. Fantastic. I will fix what I can as I go along.

Removed the definition of subtraction of natural numbers - it is not really needed.

I am converting both articles to LaTex, but I have been busy and need to catch up. Also, my LaTex is VERY rusty. So please be patient with that.

'Your second "WOLOG" paragraph under this heading jumps immediately to the conclusion that "there is a rational Q between R and R+ε, R < Q < R+ε". Why?' It follows immediately from the paragraph before, where I show there is a rational number between any two reals.

The hyperrationals are simply a way to introduce the concept of hyper-numbers. For example, while I didn't use it, you have hyperintegers that some texts also use. It shows why you need the hyperreals - it fixes the main issue with the hyperrationals. The hyperrationals can be used to define the reals as the hyperrationals infinitesimally close to each other but do not contain real numbers. The hyperreals fix this.

I do not know 'Was sind und was sollen die Zahlen?' I need to look into it and perhaps incorporate it. I learned about Dedekind Cuts and Cauchy Sequences in my first-year university subject on Analysis.

I didn't know about Fresh's article, but I will investigate it.

Regarding the links you mention, all you have to do is click logon with some of the options they give such as Facebook for Brilliant and the Medium one is free - you do not need to be a member. I don't think that is a difficulty for most people.

Thanks
Bill

 

[bhobba]

I really, really believe no. MAYBE a few people have something different to say.

I agree. I think the fundamental concept is formal set theory of some sort eg ZFC.

Thanks
Bill

 

[bhobba]

This seems to be quoted from the article https://www.revistaminerva.pt/on-the-nature-of-natural-numbers/ which looks pretty weak to me on a number of points.

Yes, I did copy it from that article. But it is WRONG. I still like the article as an interesting read.

Thanks
Bill