What is "2"?

[GrowingSmaller]

Good point q. My two cents.

 

[Architeuthus]

I find it amazing how confused you can get once you are determined to strip any frame of reference off of concepts.

Philosophers have been doing that for thousands of years.

Stripping away the "frame of reference" that you're used to is sometimes necessary for understanding.

 

[Architeuthus]

In the case of mathematics, there are a number of classical answers. Three main answers, which seem to me to be mutually exclusive, are:

There is PLATONISM. To quote G.H. Hardy, "mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations’, are simply our notes of our observations." This makes "2" an abstract thing in the Platonic world. Many great mathematicians have believed this, but critics (i) say it is quasi-religious; and (ii) wonder how this Platonic world gets "observed."

There is EMPIRICISM. Mathematics is a branch of physics, and 2 + 2 = 4 is just a generalised statement about properties of apples and atoms. This makes "2" a generalisation of pairs of real things (as some posters have suggested). Problems with this approach are: (i) what do very, very large numbers refer to? (ii) why does pure mathematics turn out to have applications to physics down the track? and (iii) why do mathematical truths seem to be necessary rather than contingent?

There is FICTIONALISM. Mathematics is a purely human construct, entirely invented. None of it is "true"; none of it relates to the real world. "2" is just a fictional construct. Problems with this are (i) why does mathematics work in science and (ii) why do animals have similar number concepts to humans?

So far I'm seeing arguments given for empiricism and fictionalism (and sometimes both at the same time).... and a lot of bemusement from people who've used numbers all their lives, but never thought about what they are.

 

[KCfromNC]

But birds have a number concept like ours.

No way to know if their internal concept is like ours or totally different. We do know that the reality they interact with is the same, so if they have a successful concept the external results will have similar successes. But understanding the problem of other minds, there's no way to know how they conceptualize the process.

So numbers must be part of reality in some sense, and not purely a human construct.

All human constructs are a part of reality, so I'm not sure what distinction you're trying to make here.

Of course we do. It's called "pure math."

So the math which applies to the real world applies to the real world and the math which doesn't, doesn't. Not sure how that helps your claims.

 

[KCfromNC]

If mathematics does that, then it cannot be purely a human construct; it must reflect some kind of reality.

Weren't you just saying something about pure math - you know, the kind which has no practical real world application?

 

[KCfromNC]

I find it amazing how confused you can get once you are determined to strip any frame of reference off of concepts.

If philosophy had any way to validate its results it wouldn't be philosophy anymore.

 

[Architeuthus]

All human construct is part of reality, so I'm not sure what distinction you're trying to make here.

I'm trying to work out if people support fictionalism or empiricism. They're inconsistent with each other, so it's a little illogical to support both.

So the math which applies to the real world applies to the real world and the math which doesn't, doesn't. Not sure how that helps your claims.

My point was that pure math often turns out to have applications to the real world long after it was invented/discovered -- but those applications had no role in the invention/discovery.

 

[KCfromNC]

Philosophers have been doing that for thousands of years.

Reminds me of this essay : How to Do Philosophy

 

[KCfromNC]

I'm trying to work out if you support fictionalism or empiricism.

Neither seems to fit my views.

I've found that much of philosophy consists of trying to stuff people into ill fitting boxes so that people can use bad arguments from the past to beat up on strawmen. This seems like a flawed approach to actually figuring things out, which may be one reason philosophy isn't particularly useful for answering questions.

The correct answer for my approach is that I support believing things which best model reality.

My point was that pure math often turns out to have applications to the real world long after it was invented/discovered -- but those applications had no role in the invention/discovery.

But when it doesn't?

 

[Architeuthus]

Neither seems to fit my views.

Well, what exactly are your views? It seems to me that numbers either have a referent in physical reality (empiricism), a referent outside physical reality (Platonism and related viewpoints), or no referent at all (fictionalism).

The correct answer for my approach is that I support believing things which best model reality.

That does sound like what I'm calling empiricism (because I don't think you're a Platonist). If not, what aspects of empiricism do you disagree with?

To talk of "modelling reality" certainly rules out the idea of "meaningless fiction."

But when it doesn't?

It happens too often to be a coincidence.

I've found that much of philosophy consists of trying to stuff people into ill fitting boxes so that people can use bad arguments from the past to beat up on strawmen. This seems like a flawed approach to actually figuring things out, which may be one reason philosophy isn't particularly useful for answering questions.

I don't care much for the implied accusation. And the boxes in this case arise from the fact that most of the coherent answers have already been provided by somebody or other in the past.

 

[True Scotsman]

But birds have a number concept like ours. So numbers must be part of reality in some sense, and not purely a human construct. The labels are unimportant (1, 2, 3, 4, 5,... or I, II, III, IV, V,... both work, as do purely mental representations).

In Introduction to Objectivist Epistemology, Ayn Rand wrote of the Crow epistemology. She was referring to a study done in the 1930's where researchers had a person walk across a meadow and into the woods. The crows in the meadow went and hid in the trees until the person walked back out of the woods and across the meadow the way he had come. When two and then 3 people walked across the meadow and into the forest the crows did the same thing, hiding until all three had left again. When 5 people crossed the meadow and into the trees the crows hid but when 4 left and 1 stayed, the crows came back out. So for crows it goes something like this: 1,2,3, Many.

 

[Resha Caner]

No, the electron is really lassoed to the proton, and the earth is really lassoed to the sun. But grouping is not real.

Saying "really" doesn't make it so. If you've got nothing else to distinguish "force" from "number" as descriptors of grouping, yet won't admit the similarity of their grouping natures, this isn't going to go anywhere.

But just so you know, it isn't a very convincing argument.

 

[Resha Caner]

I find it amazing how confused you can get once you are determined to strip any frame of reference off of concepts.

Often it's more an attempt to strip away someone else's frame of reference and substitute your own.

So what frame of reference would you propose?

 

[essentialsaltes]

If mathematics does that, then it cannot be purely a human construct; it must reflect some kind of reality.

I don't see why. When we say "That painting or that sunset is beautiful", where is the beauty happening? The painting is real, but it does not have any beauty sprinkled into the paint. The beauty is in the eye of the beholder.

 

[essentialsaltes]

Saying "really" doesn't make it so. If you've got nothing else to distinguish "force" from "number" as descriptors of grouping

Force is not a descriptor of grouping. I feel the force of gravity. I have already said that the senses give us a connection (albeit imperfect) to reality. Gravity is real. I have never felt a 2. No one has. No one can.

 

[Resha Caner]

Force is not a descriptor of grouping.

It is. When I create a set of 2 oranges, my senses have detected a property that connects them (color). When a scientist creates a set of a proton and an electron and calls it hydrogen, it is because they have detected a property that connects them (force).

I realize you think grouping oranges because of color is a human perception and grouping protons and electrons "just is", but that would not be correct. You agreed I am part of reality, so my groupings are just as "real" as the binding forces of hydrogen. And H1 (protium) is somewhat arbitrary. Why H1? Why not H2 or H3? Nature can't seem to make up its "mind".

I feel the force of gravity.

Shrug. I see the color of oranges. I don't feel the force of a single proton pulling on an electron. Does that mean it's not real?

I have never felt a 2.

I think you have. Are the binding forces of H3 unique? Why are they unique? Because there are 2 neutrons.

It's not that 2 is "attached" to the neutron, but you have experienced it. "Orange" isn't attached to the fruit. Are you going to say you haven't seen orange because it's not attached to the fruit?

 

[essentialsaltes]

It is. When I create a set of 2 oranges, my senses have detected a property that connects them (color). When a scientist creates a set of a proton and an electron and calls it hydrogen

The proton and the electron also exert forces on every other charged particle in the universe. Why not group those together? Or group together the constituents of H2O. All of them are connected by electromagnetic forces. But the boundary drawing is being done by you. The things in themselves just be.

You agreed I am part of reality, so my groupings are just as "real" as the binding forces of hydrogen.

Some descriptions of chunks of reality have real referents. Others don't.

Why H1? Why not H2 or H3? Nature can't seem to make up its "mind".

I have no idea what you're getting at. Nature is just being.

I don't feel the force of a single proton pulling on an electron. Does that mean it's not real?

We use instruments to extend our senses, and they can feel it. But you will never find a 2 in your microscope.

Are you going to say you haven't seen orange because it's not attached to the fruit?

Orange is another way we mentally describe the world. I have never seen 'naked' orange, just as I've never seen a 'naked' 2. They are both ideas. If ideas are real, then they're real. Ideas don't meet my understanding of what is real.

 

[Resha Caner]

The proton and the electron also exert forces on every other charged particle in the universe. Why not group those together? Or group together the constituents of H2O. All of them are connected by electromagnetic forces. But the boundary drawing is being done by you. The things in themselves just be.

But you said forces were real. Now you make them sound arbitrary.

Some descriptions of chunks of reality have real referents. Others don't.

The human mind is the referent. Any time you apply a specific input to the human mind - a request to a store clerk for 2 oranges - you will get a very real, consistent, and predictable result that is dependent on "2".

Orange is another way we mentally describe the world. I have never seen 'naked' orange, just as I've never seen a 'naked' 2. They are both ideas. If ideas are real, then they're real. Ideas don't meet my understanding of what is real.

Nor have you ever seen a "naked" force. That is, a "force" floating about completely apart from any material.

 

[essentialsaltes]

But you said forces were real. Now you make them sound arbitrary.

No, the groupings are arbitrary.

The human mind is the referent.

No, the referent of "cow" is that animal over there. It cannot be that the referent of 2 is the human mind. Especially if the referent of 3 is also the human mind.

Nor have you ever seen a "naked" force. That is, a "force" floating about completely apart from any material.

I have felt forces.

 

[Resha Caner]

No, the groupings are arbitrary.

So forces can group 2 electrons without employing any other matter? In my earlier use of "group", I was referring to creating a nucleus (protons & neutrons) and electron shells. But since forces allow us to group things arbitrarily ...

No, the referent of "cow" is that animal over there. It cannot be that the referent of 2 is the human mind. Especially if the referent of 3 is also the human mind.

So, given the animal is the referent for "cow", it can't be the referent for "hoof"?

And once the cow is a steak on my plate here and a steak on your plate there, are we both eating the same steak? Cuz apparently it's arbitrary to take something that was "1" and make it "2". So, since one cow was the "reality" it must be one steak - just a different name for the same reality. Right? The cow is real but the steaks are a product of my imagination.

I have felt forces.

I have seen orange.