What is "2"?

[jacknife]

Its that number that comes after 1

 

[muddleglum]

I used algebra as an analogy during a recent discussion, and was told the situations I was comparing were "apples and oranges". See here:
/#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.

"2" is a symbol that can mean anything. Certain cultures have made it mean a certain concept.

"Two" and its sound can also mean anything. Again, certain cultures have made it mean a certain concept.

That concept seems to be wired in, if my memory of neurophysiology is correct. There are number systems that cannot conceptualize the concept. :-D For instance, an Abelian ring that just has a multiplicative identifier and additive identifier. (BTW, it has been years since I've taken abstract algebra so don't assume I remember the correct term for everything.)

As far as that apples and oranges response is concerned; as you do, I don't understand it from that person's perspective. BTW, I usually talk about partial derivatives so that I can demonstrate the steps one has to take to really understand certain concepts. (OTOH, I think I could say that it is an apples/oranges response in one sense from your Christian perspective, if I understand your reasoning correctly.)

On the "sacraments" you seemed to have come to the same conclusion I have and I only attended one Lutheran church in my life and have no idea what was said (I'm non-hearing). I find it interesting that wherever I go, there are those who have come from quite different backgrounds that believe the same thing, though often with different terminology.
If one listens, they can hear the similarity, but if one wants to quibble, and I love to quibble, they can make the differences dwarf the Grand Canyon.

 

[essentialsaltes]

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?

Natural numbers are basically defined by the Peano axioms I linked to before. They are all assumed in this system.

0 is assumed to be a natural number. (Or, if you like, 0 is the arbitrary name of a natural number.)

We assume every natural number has a successor. That's all the definition you need for succession. 0 has a successor (by assumption) and we typically call it 1. 2 is the successor of 1.

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

It is arbitrary in the sense that we could have assumed different axioms to set up our arithmetic. And obviously we could have chosen a different name than "2".

 

[essentialsaltes]

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.

Those are not definitions, those are just different names for the same thing.

 

[essentialsaltes]

But what is 2? Is it an abstract Platonic concept?

It is an idea. We can use ideas to describe the real things we see out in the world. But this is not a representation of an idea, but the use of an idea as a description.

You may call the thing under this text a line, but it clearly is not infinitely long with no thickness, as a mathematical line is. You can call the picture of the two apples a 2, or a representation of a 2, but that's sloppy usage, if we're talking about mathematics.

The picture of the 2 apples does not have any 2 in it. When we describe the picture, we could describe it as a picture of 2 apples. Or a picture of 1 pair of apples. Or a picture composed of 15636 pixels, depending on what aspect of the picture we were interested in describing with the ideas we call numbers.

 

[Resha Caner]

It is arbitrary in the sense that we could have assumed different axioms to set up our arithmetic.

It is an idea. We can use ideas to describe the real things we see out in the world. But this is not a representation of an idea, but the use of an idea as a description.

As I recall, you don't like to debate the philosophy of such things. So I'm not sure if I should ask any follow-on questions of you. If I did, they would be:

1) If we began our number system with different assumptions, would we produce different results in the "real" world by using that different system?

2) If we're using numbers to describe real things - what I would call a "property" of those real things - are we describing something inherent to these real things?

 

[Cearbhall]

I used algebra as an analogy during a recent discussion, and was told the situations I was comparing were "apples and oranges".

Looks like they were.

 

[quatona]

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.

"2" in algebra (as an entirely abstract concept), or "2" in "2 [insert object description of choice]"?
I´m asking because in the discussion so far you used the two interchangeably, which caused your confusion.

 

[essentialsaltes]

1) If we began our number system with different assumptions, would we produce different results in the "real" world by using that different system?

Making different assumptions doesn't change the real world. Different mathematical systems are better or worse ways to describe various parts of the real world. If the time is described as eight, and you wait six hours, in some parts of the world it will be described as two, and in other parts it will be described as fourteen. Different mathematical systems give different descriptions, but the underlying reality is the same in both cases.

2) If we're using numbers to describe real things - what I would call a "property" of those real things - are we describing something inherent to these real things?

Not necessarily. If I correctly describe that penny as the sixth penny I've picked up from the sidewalk this week, there is nothing inherently sixy about the penny.

 

[Resha Caner]

Different mathematical systems give different descriptions, but the underlying reality is the same in both cases.

So language may be fluid, but "2" is an attempt to describe something "real" about the world. Yes?

Not necessarily. If I correctly describe that penny as the sixth penny I've picked up from the sidewalk this week, there is nothing inherently sixy about the penny.

Whoa. "6". Let's not get fancy here. Can we talk about penny 2? My head might explode if I have to go all the way to "6".

So, while there isn't something "twoish" about a particular penny, isn't there something "twoish" about the set? ... It is interesting the property we're discussing seems to appear with systems and disappear when the system is broken up ... it's almost emergent. But again, we don't want to get crazy about this stuff.

 

[Architeuthus]

In this case, "2" being a concept of quantity, it is the quantity that is the essential

Is "concept of quantity" the same as "non-negative integer"?

The concept "2" subsumes all instances of the quantity 2 of anything be it apples, mountains or unicorns.

Yes, but what is the concept "2"? And how does it "subsume all instances of the quantity 2 of anything"?

 

[Architeuthus]

Those are not definitions, those are just different names for the same thing.

Those were two possible definitions of things that satisfied the axioms, one a reasonably standard one in terms of set theory.

Now, you might say (and I might agree with you) that anything satisfying the Peano axioms "works," and so we don't need a precise definition of "2."

However, anybody that doesn't like a Platonic "2" concept is kind of obliged to define "2" more precisely, and some have tried.

 

[essentialsaltes]

So language may be fluid

Not the language, but the mathematics can be whatever you assume it to be.

but "2" is an attempt to describe something "real" about the world. Yes?

2 is an idea. We often use it as an analogy to describe things we see in the real world. Like lines on a paper are only a poor analogy for Euclidean lines.

So, while there isn't something "twoish" about a particular penny, isn't there something "twoish" about the set?

I don't think so. You're throwing your mental construct (an idea) onto some chunk of reality. But it's not in the reality, it's in your head.

It is interesting the property we're discussing seems to appear with systems and disappear when the system is broken up.

How so? In 1982, 16,725,504,368 pennies were minted. They have all been dispersed, and some destroyed, and yet this number remains behind unaffected to describe the pennies of that year.

When the first one was minted, was there an inherent association between it and 16,725,504,368? What if one was circulated, but got melted down before the last one was minted in 1982? There could never have been all of them assembled at once. But that has no effect on the quantity associated with a category that is created in our heads for pennies minted in a particular year.

Like the sixth penny I found on the ground this week. Two of the earlier ones had already been spent by the time I found the sixth one. The six pennies were never together, except in my mind. My mind associates them together, and can describe them. There is no inherent association of these objects.

 

[essentialsaltes]

Those were two possible definitions of things that satisfied the axioms, one a reasonably standard one in terms of set theory.

I disagree, they were like translations into different languages, not novel definitions. Identifying 0 with {} is just a translation.

The axioms define the logical structure of the system. If you accept the axioms, the successor of the successor of the first element is the answer to the OP. No matter how you choose to write it.

 

[Architeuthus]

It is an idea. We can use ideas to describe the real things we see out in the world. But this is not a representation of an idea, but the use of an idea as a description.

If you mean "idea" in a Platonic sense, I would agree.

If you mean "idea" in a subjective sense, I would not.

 

[Architeuthus]

I disagree, they were like translations into different languages, not novel definitions. Identifying 0 with {} is just a translation.

I think we're using "define" in different senses. If it was "just a translation" there would not have been so much published on the question. Either numbers are really special kinds of sets, or they are not (personally, I would say not).

The axioms define the logical structure of the system.

They define the structure of the system in that all definitions satisfying the Peano axioms are isomorphic (although not identical).

 

[Resha Caner]

I don't think so. You're throwing your mental construct (an idea) onto some chunk of reality. But it's not in the reality, it's in your head.

What else have we got? That's how we associate with the world. Just because I call a leaf on a tree "green" and associate it with a concept of "color" doesn't mean the leaf is not reflecting light at specific wavelengths.

Speaking of light - there's a number we can't change: 3E8.

 

[essentialsaltes]

What else have we got?

Maybe nothing, but we shouldn't mistake what's in our heads for what's out there in reality.

ust because I call a leaf on a tree "green" and associate it with a concept of "color" doesn't mean the leaf is not reflecting light at specific wavelengths.

I think this is a separate issue, but.... there's the spectral reflectance of vegetation. It reflects more NIR than it does 'green'. And it reflects a continuous band of wavelengths, not just 'specific wavelengths'. Maybe one of those NIR wavelengths is the color it really is.

Or is the leaf really green because your eye is blind to most of what it's reflecting?

 

[Resha Caner]

Maybe nothing, but we shouldn't mistake what's in our heads for what's out there in reality.

Agreed. Instrumentalism has long been my view. Yet for some reason you seem to believe there is something "out there" that is "reality". How did you come to that conclusion if not by forming an idea from your perceptions?

Are you wrong that reality is out there?

I think this is a separate issue, but.... there's the spectral reflectance of vegetation. It reflects more NIR than it does 'green'. And it reflects a continuous band of wavelengths, not just 'specific wavelengths'. Maybe one of those NIR wavelengths is the color it really is.

Or is the leaf really green because your eye is blind to most of what it's reflecting?

This would be moving the goal posts. The concept of color was never a claim to perceive all electromagnetic radiation reflected by the leaf - only "specific wavelengths" that are known as green within the concept of color. Are you denying a leaf reflects those wavelengths?

And this is still related to number. I still await a comment on 3E8. I know it's not "2", but hopefully it's still sexy enough to talk about that number as well.

 

[TillICollapse]

Thinking out loud:

Language which we use to communicate ideas and concepts concerning, is arguably a public and social phenomenon, and so it's applicable in public and social context. A language that is only understood privately, is essentially irrelevant and incoherent. I think that may be why, when a person tries to definitively nail down a definition of a thing for the use of communication, but in doing so refers to the qualia of the self only ... they come up short, and it's because it cannot be put into words which only the self will understand. Thus, it must be something we agree upon publicly and socially per communication purposes. That doesn't change whatever the thing which the language is being mapped to ACTUALLY is or not ... only that attempts to refer to private experience only to define it via communicative language, comes up short.

IOW, what "2" is may be discovered privately, but cannot be communicated privately.

Thoughts ?