What is "2"?

[essentialsaltes]

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?

It's possible, but I don't believe that. I assume reality exists. I assume my senses give me an (imperfect) way to gain knowledge about it. But I don't want to mistake my description of reality for reality.

Are you denying a leaf reflects those wavelengths?

No, but it also reflects red and blue wavelengths as well. Why isn't it red? Or blue? You could use 3 numbers to describe the RGB values. Or 4 if you add brightness. How would you like to describe the leaf with your mental ideas? The leaf is the same, but you can describe it with different mental schema. The picture of two apples is 1 picture of 2 apples composed of 64 colors and 12936 pixels. What part of it do you want to describe? You can use numbers to accurately describe reality, but the correspondence or association between things and numbers is all in your head. Because that's what numbers are. Ideas in your head.

Numbers are the map, not the territory.

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.

This may surprise you, but meters and seconds are also things we invented to describe the world. I'm not sure what comment you want me to make on 3E8. If we describe things in meters and seconds, then that's roughly the quantity that correctly describes the thing we have defined as velocity for light in vacuum.

I can change the raw number to 186,000 or 1 quite easily by describing the universe in different manmade units.

 

[True Scotsman]

Is "concept of quantity" the same as "non-negative integer"?[Quote/]

There are different types of concepts. There are concepts of action such as fishing and archery. There are concepts of consciousness such as, well, concept or feeling. The reason we need concepts of quantity is that our brains can deal with only a very limited number of units (members of a group of similar objects) at a time. For Humans it's about 6. Ever noticed that when a sentence is very long and contains a lot of concepts that you have trouble keeping track? That's why.


If I type a number of slash marks like this: //////////////////////////////////, without counting them you can't do anything with them. But, If I type 34, you can. Numbers are symbols that stand for a concept of a specific quantity. If I type the number 48 you instantly know what I mean just as if I type the word sanswich you instantly know what I'm referring to. Just imagine doing math without them.

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

A concept is it's concretes, not its definition. So the concept 2 means any instance of more than one unit and less than 3 units, or two units. In the case of the concept two, the concretes can be any unit but must be some unit, all other attributes are omitted and the essential characteristic, the quantity, is retained. Just as the concept "man" subsumes all men, past, present and future, the concept two subsumes all instances of the quantity two.[QUOTE/]

 

[Architeuthus]

Numbers are the map, not the territory.

That seems to be a disagreement with the Platonist philosophy of mathematics. I would tend to think that, in mathematics, numbers are the territory.

 

[Architeuthus]

There are different types of concepts. There are concepts of action such as fishing and archery. There are concepts of consciousness such as, well, concept or feeling.

That seems to be suggesting that numbers are subjective things.

A concept is it's concretes, not its definition. So the concept 2 means any instance of more than one unit and less than 3 units, or two units.

I can't even begin to understand what "a concept is it's concretes, not its definition" means.

Do you mean "the concept 2 means the set of instances of more than one unit and less than 3 units"?

And what about concepts for very large numbers, like 10^100? Sufficiently large numbers have no corresponding concrete instances.

 

[Architeuthus]

Let's recap. Amidst all the confusion, I can see four possible answers for what "2" might be.

a) A Platonic number concept; specifically, the one given by the expressions 1 + 1 or 7 - 5.

b) S(S(0)), in some system satisfying the Peano axioms (it might not matter which). This is consistent with (a), or with attempts to define numbers in terms of set theory (as in Principia Mathematica, but see also What Numbers Could Not Be). The Peano axioms imply that there are infinitely many non-negative integers.

c) Some kind of purely subjective idea or concept. This is inconsistent with (a) and (b), and raises serious questions about why mathematics works and how mathematical concepts can be shared between people. It also implies that there can be only finitely many non-negative integers (although the set of non-negative integers may contain "gaps" for integers that nobody has ever thought about).

d) Some kind of collection (set?) of all possible pairs of real, physical things, such as the pair of apples below. This is inconsistent with all of the above. This option also implies that there can be only finitely many non-negative integers, and that integers above some threshold N do not exist.

Maybe we could have a poll or something?

 

[TillICollapse]

Let's recap. Amidst all the confusion, I can see four possible answers for what "2" might be.

a) A Platonic number concept; specifically, the one given by the expressions 1 + 1 or 7 - 5.

b) S(S(0)), in some system satisfying the Peano axioms (it might not matter which). This is consistent with (a), or with attempts to define numbers in terms of set theory (as in Principia Mathematica, but see also What Numbers Could Not Be). The Peano axioms imply that there are infinitely many non-negative integers.

c) Some kind of purely subjective idea or concept. This is inconsistent with (a) and (b), and raises serious questions about why mathematics works and how mathematical concepts can be shared between people. It also implies that there can be only finitely many non-negative integers (although the set of non-negative integers may contain "gaps" for integers that nobody has ever thought about).

d) Some kind of collection of all possible pairs of real, physical things, such as the pair of apples below. This is inconsistent with all of the above. This option also implies that there can be only finitely many non-negative integers, and that integers above some threshold N do not exist.

Maybe we could have a poll or something?

In your opinion, do any of your four possible answers address or factor in what "2" might be (or any number for that matter) as it concerns animals which apparently demonstrate math ability but do not communicate what "2" is ?

 

[Architeuthus]

In your opinion, do any of your four possible answers address or factor in what "2" might be (or any number for that matter) as it concerns animals which apparently demonstrate math ability but do not communicate what "2" is ?

See also More Animals Seem to Have Some Ability to Count - Scientific American

I think that all the options except (c) allow for explanations of animals being able to count.

 

[True Scotsman]

That seems to be suggesting that numbers are subjective things.

They are mental integrations with objective meaning.

I can't even begin to understand what "a concept is it's concretes, not its definition" means.

That's OK. You've probably never heard anything like it before and I'm probably doing a bad job of explaining it. If you are interested I suggest you read An Introduction to Objectivist Epistemology.

Do you mean "the concept 2 means the set of instances of more than one unit and less than 3 units"?

Yes. Just as the concept man means the set of all men.

And what about concepts for very large numbers, like 10^100? Sufficiently large numbers have no corresponding concrete instances.

Well that's right. They are only potential or the continuation of a sequence. The concept "infinity" has no objective reference. It means a number bigger than any number you could imagine. But we were talking about the concept "2".

 

[TillICollapse]

See also More Animals Seem to Have Some Ability to Count - Scientific American

I think that all the options except (c) allow for explanations of animals being able to count.

Hmm.

Well I wasn't asking if they allowed for explanations for animals being able to count, rather, when trying to identify what "2" is for example, did they factor in what animals demonstrate.

Perhaps a more focussed example:

Would it be possible to deduce what an ant would recognize as "2" by observing it walk and navigate while it arguably counts it's steps ? Why or why not ?

 

[TillICollapse]

They are mental integrations with objective meaning.




That's OK. You've probably never heard anything like it before and I'm probably doing a bad job of explaining it. If you are interested I suggest you read An Introduction to Objectivist Epistemology.

Yes. Just as the concept man means the set of all men.



Well that's right. They are only potential or the continuation of a sequence. The concept "infinity" has no objective reference. It means a number bigger than any number you could imagine. But we were talking about the concept "2".

Out of curiosity ... at what age (yours) did you first start to find value in Rand's philosophies/perspectives/etc (I'm assuming) ? If you'd rather not answer, no worries. Just curious.

 

[Architeuthus]

when trying to identify what "2" is for example, did they factor in what animals demonstrate.

Who is "they" here?

 

[TillICollapse]

Who is "they" here?

"They" were your 4 possibilities. Sorry.

Actually, I'm more interested in possible answers to this:

Would it be possible to deduce what an ant would recognize as "2" by observing it walk and navigate while it arguably counts it's steps ? Why or why not ?

 

[True Scotsman]

Let's recap. Amidst all the confusion, I can see four possible answers for what "2" might be.

a) A Platonic number concept; specifically, the one given by the expressions 1 + 1 or 7 - 5.

b) S(S(0)), in some system satisfying the Peano axioms (it might not matter which). This is consistent with (a), or with attempts to define numbers in terms of set theory (as in Principia Mathematica, but see also What Numbers Could Not Be). The Peano axioms imply that there are infinitely many non-negative integers.

c) Some kind of purely subjective idea or concept. This is inconsistent with (a) and (b), and raises serious questions about why mathematics works and how mathematical concepts can be shared between people. It also implies that there can be only finitely many non-negative integers (although the set of non-negative integers may contain "gaps" for integers that nobody has ever thought about).

d) Some kind of collection (set?) of all possible pairs of real, physical things, such as the pair of apples below. This is inconsistent with all of the above. This option also implies that there can be only finitely many non-negative integers, and that integers above some threshold N do not exist.

Maybe we could have a poll or something?

I think you are making this much more difficult than it needs to be. This is a simple, simple concept. If you are having a hard time with "2" then how can you define a god?

 

[Architeuthus]

Yes. Just as the concept man means the set of all men.

Well, why didn't you say so?

Well that's right. They are only potential or the continuation of a sequence. The concept "infinity" has no objective reference. It means a number bigger than any number you could imagine. But we were talking about the concept "2".

It follows from the "Objectivist" approach that statements about sufficiently large numbers are meaningless. If, say, 2^43112609−1 is over the threshold of numbers that have an objective reference, then mathematicians are talking nonsense when they say it's prime. Similarly all statements about "infinity" would be nonsense, as would statements about aleph numbers. That's a fair chunk of mathematics being thrown out of the window.

 

[Architeuthus]

"They" were your 4 possibilities. Sorry.

Like I said, animal mathematics is only problematic for (c), I think. It would also be problematic for (e), the idea that mathematics is a cultural construct (since animals aren't part of human culture).

Actually, I'm more interested in possible answers to this:

Would it be possible to deduce what an ant would recognize as "2" by observing it walk and navigate while it arguably counts it's steps ? Why or why not ?

I don't know much about ants counting, I'm afraid (if they indeed do count). It does sound intriguing, though.

Birds seem to either subitise or to count up to some very small limit.

 

[Architeuthus]

I think you are making this much more difficult than it needs to be. This is a simple, simple concept.

This is the philosophy forum, right?

The foundations of mathematics are actually not quite so obvious as they seem to be. Especially since we all recognise "2" when we see an instance of it, and we haven't struggled with small integers since kindergarten.

But if it was simple, we would all be able to agree on what exactly "2" was. And Russell and Whitehead wouldn't have needed more than 300 pages to prove that 1 + 1 = 2:

And I don't get how you can combine "it's simple" with a radical Randian approach that would toss a lot of accepted mathematics away.

 

[TillICollapse]

I don't know much about ants counting, I'm afraid (if they indeed do count). It does sound intriguing, though.

Birds seem to either subitise or to count up to some very small limit.

Okay, so let's focus on the chicks. They apparently utilize what we recognize as "2" ... would you agree ? Take for example:

Take the domestic chicken (Gallus gallus), a bird that many think of as having more to do with barbecue sauce than with arithmetic. If a chicken sits in front of two small opaque screens, and one ball disappears behind the first screen, followed by four balls disappearing behind a second screen, the chicken walks towards the screen that hides four balls, since four balls are better than one ball. The feat is made more impressive when you consider that the chicken in question is only three days old.

And it can do a lot more than add up.If one ball disappears behind the first screen, and four balls disappear behind the second, just as before, but then two of the four balls behind the second screen are visibly moved over to the first screen, the chicken is now faced with two tasks. It must add two to one, and know that there are now three balls behind the first screen. It must also subtract two from four, and realise that there are only two balls left behind the second screen. The young chicken must overcome its initial impulse to approach the second screen, which initially hid four balls, and instead approach the first screen, now hiding three balls. If this sounds complicated for the three-day-old bird, think again. Infant chickens correctly approached the screen hiding more balls nearly 80% of the time.

So what is "2" from the perspective of the chicken utilizing it ? Is there a way we could know what the chicken identifies as "2" from it's own perspective ... why or why not ?

 

[True Scotsman]

Out of curiosity ... at what age (yours) did you first start to find value in Rand's philosophies/perspectives/etc (I'm assuming) ? If you'd rather not answer, no worries. Just curious.

I don't mind at all. I was 27 or 28 at the time. I had no philosophy to speak of. Just a jumble of un-integrated ideas and contradiction. I was trying to be a Christian. My Grandmother gave me a copy of Atlas Shrugged to read. At first I hated it. It went against everything I had been taught. I was ranting and arguing with it the whole time. It was all bent and torn from me throwing it against the wall. I didn't finish it then but it nagged at me. I read it again a couple of years later and vowed that I would refute everything in it. There was just one problem: I couldn't.

By the time I finished it I was convinced. I then read Anthem, The Fountain Head and We the Living. I determined then and there that I would learn this philosophy backwards and forwards. I read all of her non-fiction works. I didn't just read them, I studied them. I started to read other philosophers and compared. It was really the non-fiction that helped me the most. I also listened to a course given by Leonard Piekoff called Objectivism through Induction and that was when I really started to understand.

I have all of her works as well as many others on philosophy on my ipod. I listen daily and I am always studying. I am a woodcarver and it affords me plenty of time to listen and integrate these ideas.

It has changed my life. I no longer feel like the world is incomprehensible. I know longer feel that fear and guilt that I used to. It is the most life affirming thing I have ever encountered.

If you have never read any of her works I would recommend Philosophy, Who Needs It as sort of an introduction.

 

[Architeuthus]

Okay, so let's focus on the chicks. They apparently utilize what we recognize as "2" ... would you agree ?

Yes.

So what is "2" from the perspective of the chicken utilizing it ? Is there a way we could know what the chicken identifies as "2" from it's own perspective ... why or why not ?

I don't know how you could, although whatever it does seems to be consistent with our "2" (or, with the universal "2," if one believes in that).

With large mammals, brain scans etc. have shed some light on cognition, but (as Wittgenstein said), "If a lion could speak, we could not understand him."

 

[TillICollapse]

I don't mind at all. I was 27 or 28 at the time. I had no philosophy to speak of. Just a jumble of un-integrated ideas and contradiction. I was trying to be a Christian. My Grandmother gave me a copy of Atlas Shrugged to read. At first I hated it. It went against everything I had been taught. I was ranting and arguing with it the whole time. It was all bent and torn from me throwing it against the wall. I didn't finish it then but it nagged at me. I read it again a couple of years later and vowed that I would refute everything in it. There was just one problem: I couldn't.

By the time I finished it I was convinced. I then read Anthem, The Fountain Head and We the Living. I determined then and there that I would learn this philosophy backwards and forwards. I read all of her non-fiction works. I didn't just read them, I studied them. I started to read other philosophers and compared. It was really the non-fiction that helped me the most. I also listened to a course given by Leonard Piekoff called Objectivism through Induction and that was when I really started to understand.

I have all of her works as well as many others on philosophy on my ipod. I listen daily and I am always studying. I am a woodcarver and it affords me plenty of time to listen and integrate these ideas.

That's very interesting ... thanks for answering. Interesting to read the impact it had on you

It has changed my life. I no longer feel like the world is incomprehensible. I know longer feel that fear and guilt that I used to. It is the most life affirming thing I have ever encountered.

If you have never read any of her works I would recommend Philosophy, Who Needs It as sort of an introduction.

Oh I've read some of her books I would probably say The Fountainhead (read when I was 17) is one of the best novels I have ever read.