What is "2"?

[Resha Caner]

I used algebra as an analogy during a recent discussion, and was told the situations I was comparing were "apples and oranges". See here:
http://www.christianforums.com/t7877904-25/#post67475629

Given my poor, addled, foolish Christian brain, I don't understand all I should about higher mathematics. So, I don't understand how fruit and the arithmetic mentioned in the video were supposed to be an answer regarding my analogy about algebra.

So let's start with this: Can someone please define "2" for me. Thanks. This is really a puzzler.

 

[Architeuthus]

Can someone please define "2" for me. Thanks. This is really a puzzler.

This will be fun. While they're at it, can they define π and i too, please?

 

[essentialsaltes]

So let's start with this: Can someone please define "2" for me. Thanks. This is really a puzzler.

Under the Peano axioms of standard arithmetic of the natural numbers, 2 is the successor of the successor of 0, assumed to be the first natural number. (Or, if you like, "2" is the name we assign to the successor of the successor of 0. "2" is the name assigned to S(S(0)).

Under other assumptions, 2 plays an analogous role in the integers, real numbers, complex numbers, etc.

 

[Resha Caner]

Under the Peano axioms of standard arithmetic of the natural numbers, 2 is the successor of the successor of 0, assumed to be the first natural number. (Or, if you like, "2" is the name we assign to the successor of the successor of 0. "2" is the name assigned to S(S(0)).

So in order to understand "2", I need to understand "natural number", "0", and "successor" (is that the same as S?). Could you define those please?

And does this mean that "2" is an arbitrary definition? That it could easily be something else if one preferred?

 

[Architeuthus]

Under the Peano axioms of standard arithmetic of the natural numbers, 2 is the successor of the successor of 0, assumed to be the first natural number. (Or, if you like, "2" is the name we assign to the successor of the successor of 0. "2" is the name assigned to S(S(0)).

Under other assumptions, 2 plays an analogous role in the integers, real numbers, complex numbers, etc.

But you're not actually defining 2 there, you're just saying it's something that satisfies the Peano axioms.

If I define 0 = {} and S(x) = {x}, then 2 would be {{{}}}

If I define 0 = (blank) and S(x) = | next to x, then 2 would be ||

Many other definitions of 2 would work equally well.

 

[Eudaimonist]

Something and something else, selected mentally.


eudaimonia,

Mark

 

[Resha Caner]

Something and something else, selected mentally.

Are you proposing this as your definition? So "2" is subjective - dependent upon what I select?

I mentally select everything as the "something" and my left eye as the "something else" and that's "2"? Interesting.

 

[GrimKingGrim]

Looking for the point

 

[Eudaimonist]

Are you proposing this as your definition? So "2" is subjective - dependent upon what I select?

Of course. "2" has no meaning otherwise.

Of course, you can have an abstract something and something else in which you don't specify specific entities, but they are assumed.


eudaimonia,

Mark

 

[Architeuthus]

Something and something else, selected mentally.

Well, this is an instance of 2, in some sense, but not a definition of 2:

 

[Eudaimonist]

Well, this is an instance of 2, in some sense, but not a definition of 2:

It is an example that explains the abstract principle. (They don't have to be apples, and I didn't insist on apples.) That's enough.


eudaimonia,

Mark

 

[Resha Caner]

Well, this is an instance of 2, in some sense, but not a definition of 2:

It does seem mathematics is always about fruit in one way or another.

Of course. "2" has no meaning otherwise.


Of course, you can have an abstract something and something else in which you don't specify specific entities, but they are assumed.

This is fun. Let's try another example. I'll select the fruit shown above as the "something" and another piece of fruit as the "something else".

So this is "2"?

It seems "2" can be anything I want it to be. How is this a useful concept?

 

[Architeuthus]

It is an example that explains the abstract principle. (They don't have to be apples, and I didn't insist on apples.) That's enough.

It presupposes that you have the abstract principles 0, 1, 2, 3, 4, ... in your head already, and identifies the third concept in that sequence.

I'm not sure it explains anything at all. Even with small children, I'm not sure if they learn what 2 means, or if the concept is "wired in" in some way and it just needs a word. Certainly as adults we simply recognise instances of "2" without much thought at all.

But Resha Caner asked a much deeper question: what actually is 2?

 

[HitchSlap]

Well, this is an instance of 2, in some sense, but not a definition of 2:

two
to͞o/Submit
number
equivalent to the sum of one and one; one less than three; 2.
"two years ago"
synonyms: pair, duo, duet, double, dyad, duplet, tandem; archaictwain
"the last two on the dance floor will win the grand prize"
a group or unit of two people or things.
noun: two; plural noun: twos
"they would straggle home in ones and twos"
synonyms: pair, duo, duet, double, dyad, duplet, tandem; archaictwain
"the last two on the dance floor will win the grand prize"
two years old.
"he is only two"

 

[Eudaimonist]

This is fun. Let's try another example. I'll select the fruit shown above as the "something" and another piece of fruit as the "something else".

So this is "2"?

Yes, it is. It can be as long as you are clear that it is two collections of apples, in which one collection has one apple and the other collection has two apples. That isn't indicated in the drawing, however.

If you wish, I can stipulate that the "something" and "something else" must be in the same conceptual category, even if that is "entity". The reason I didn't mention that is that I was speaking as generally as I could.

It seems "2" can be anything I want it to be. How is this a useful concept?

It is useful as long as you are clear in what you are counting. You can also count an apple and an orange as long as you are clear that you are counting fruit, and not apples.

As long as you are not intentionally playing dense, you should be able to find counting very useful.


eudaimonia,

Mark

 

[Architeuthus]

two
to͞o/Submit
number
equivalent to the sum of one and one; one less than three; 2.
"two years ago"
synonyms: pair, duo, duet, double, dyad, duplet, tandem; archaictwain
"the last two on the dance floor will win the grand prize"
a group or unit of two people or things.
noun: two; plural noun: twos
"they would straggle home in ones and twos"
synonyms: pair, duo, duet, double, dyad, duplet, tandem; archaictwain
"the last two on the dance floor will win the grand prize"
two years old.
"he is only two"

The closest thing to a definition there is "equivalent to the sum of one and one," which boils down to the same as the S(S(0)) mentioned earlier, given that S(0) + S(0) = S(0 + S(0)) = S(S(0)).

But what is 2? Is it an abstract Platonic concept? Is it a set? (and if so, which set?) Is it a process of some kind? (e.g. a counting process?) Is it something else?

I'm sure we can all recognise an instance of "2" when we see it, but what exactly is "2"?

 

[True Scotsman]

Well, this is an instance of 2, in some sense, but not a definition of 2:

This is where an objective theory of concepts is crucial. What is a definition? According to the OTC it is a description of the concretes subsumed by a concept and it is the essentials that are included in the definition. In this case, "2" being a concept of quantity, it is the quantity that is the essential The concept "2" subsumes all instances of the quantity 2 of anything be it apples, mountains or unicorns.

 

[Resha Caner]

Yes, it is. It can be as long as you are clear that it is two collections of apples, in which one collection has one apple and the other collection has two apples. That isn't indicated in the drawing, however.

It seems a bit circular that I would indicate this represents "2" by indicating that the collection on the right is also "2".

It also seems my presentation of the apples was not clearly "2". Is there another possible interpretation?

 

[Tinker Grey]

You gotta keep your units straight. A knife and a fork are two utensiles or two objects but they are not two knives, two forks, nor two apples.

 

[HitchSlap]

The closest thing to a definition there is "equivalent to the sum of one and one," which boils down to the same as the S(S(0)) mentioned earlier, given that S(0) + S(0) = S(0 + S(0)) = S(S(0)).

But what is 2? Is it an abstract Platonic concept? Is it a set? (and if so, which set?) Is it a process of some kind? (e.g. a counting process?) Is it something else?

I'm sure we can all recognise an instance of "2" when we see it, but what exactly is "2"?

Two is exactly that thing we mean when you're two dollars short of bus fare, and not one.