Are numbers as real as rocks?

[weltweit]

That's the wider question of ontology.

But we can recognise things as real, or discuss why we think of this or that as real or not, or in what ways, even while being unable exactly to define what we mean by "real".

Interesting link, thanks.

Well ok .. for me then rocks and everything in the physical world and all the clutter of my life which is around me right now is real, it is in my reality.

Argument ideas and academia of various types, because they are man made and sometimes abstract are less real for me than the physical world. After all many of them were only invented to explain the physical world slightly better than a previous explanation which they replaced.

What I can see with my eyes and feel with my senses is, to me, the most real.

 

[muser]

Yes, but that speed of light analogy shows that you do not have to be able to kick something for it to be real. Imo. Do you disagree

Wasn't newton's aether theory presumed correct for several centuries before Einstein. Michele besso and einstein used to discuss scientific matters on the way home from his patent office. It's purported that he (einstein) put the rudimentary ideas of SR to Besso in order to gain a engineering (real world) perspective. thereby testing his more salient ideas and discarding others.
Einstein learnt most of the mathematical constructs, for his theories, at a later stage, and then only to give his paper, crediblity among his peers.
It could be argued that maths is used to decipher some further truth when the main body has been discovered.

 

[Knotted]

I mean, the splashes of yellow are a bowl of fruit in the story of the other colours (just like 1+1=2 is true the story of mathematics). Now, in the still life painting a fiction (that those splashes of colour are a bowl of fruit) is true. Reasoning by analogy you can say that it is possible that the fiction of 1+1=2 is true in sceince. This is correct, no?

This is close to what I'm saying. I don't think 1+1=2 is true in science though it might be true if we talk concretely about what the 1's and the 2 represent.

Importnatly, the painting and sceince are as real as each other, or you are just pointing out that something that is fictional can be true in a fiction - not very useful. So that I conclude that you are talking about the brute physical picture, not an abstraction, or you be bizzarely conceding that although mathematics is not real, the idea of a picture, is.

The second one. Although I would say mathematical formality rather than mathematics.

Now comes my inspiration:
Now, the way I see it is that that these splashes of colour are a bowl of fruit in the patinting, must be an illusion. That is quite intuitive, especially considering that you do say that there is no bowl of fruit. So I conclude that maths is true in science as an illusion, or the analogy is flawed.
Thats my reasoning. Please explain if its nothing like what you meant.

Yes, I like that. Although my use of the analogy of the painting was supposed to make a distinction between accurracy and usefulness rather than a distinction between accurracy and realism. So I didn't talk about the painting being an illusion nor did I talk about the 'truth of the painting' if that even makes sense.

I mean: How can x be y? The same way that A is B. The splashes of colour are true in the painting by illusion (or how else - I assume that this is where we disagree). If you have answered the question, then x must be y by an illusion (if you agree that a splasges of colour are true in a painting by illusion), otherwise you have not answered the question and gained no credibility to the view that it is possible, as the analogy is flawed at the very root.

Well firstly I don't claim to have answered the question. If anything I was trying to show that the question is a very difficult one. If I could answer it in one week then I'd do the rest of philosophy in a day.

I think that the statement x=y is either a tautology like, "pegasus can fly so pegasus can not not fly" or it is a statement like, "for our purposes we say that x and y are interchangeable." You could call the latter an illusion I suppose, although abstraction is better in my view.

 

[118118]

Hello Knotted. Can you run past me one more time, what evidence/argument you have for 1+1=2 not being objectvely true. As is demonstrated in my last post, I always misunderstand what you are saying. Or can anyone else explain what you are saying? I mean, explicitly can you show the working out, or structure of the argument? If not, then I will not say that you are right, but there is no way that I can disagree

 

[muser]

Hello Knotted. Can you run past me one more time, what evidence/argument you have for 1+1=2 not being objectvely true. As is demonstrated in my last post, I always misunderstand what you are saying. Or can anyone else explain what you are saying? I mean, explicitly can you show the working out, or structure of the argument? If not, then I will not say that you are right, but there is no way that I can disagree

reread post 84.

 

[118118]

Ok, if thats it. I don't think that knotted can be claiming anything other than fictional truth to 1+1=1. It is an objective truth that pegasus does not exist. It is a fictional truth that pegasus does. The truth in the story of Greek mythology has no bearing on the objectivity of the claim that pegasus does not exist. Objective facts can exist even though they are contradicted by fictional ones.

I assume that I've misunderstood knotted's chain of reasoning, however.

 

[muser]

Ok, if thats it. I don't think that knotted can be claiming anything other than fictional truth to 1+1=1. It is an objective truth that pegasus does not exist. It is a fictional truth that pegasus does. The truth in the story of Greek mythology has no bearing on the objectivity of the claim that pegasus does not exist. Objective facts can exist even though they are contradicted by fictional ones.

I assume that I've misunderstood knotted's chain of reasoning, however.

1 apple + 1 pear = 1 apple. If you think of it in those terms then maybe it makes more sense. 1 + 1 = 2, doesn't represent both of those fruit mentioned. Do you agree?

 

[118118]

1 apple + 1 pear = 1 apple. If you think of it in those terms then maybe it makes more sense. 1 + 1 = 2, doesn't represent both of those fruit mentioned. Do you agree?

Maths could easily represent what is going on there. If maths could not function in models representing plastecine, then it would be of very limited use in science.

So did I misundersatnd the reasoning behind it in my last post? - I mean Knotted cannot be claiming anything other than fictional stauts to 1+1=1.

Perhaps it is that no objectively true system could treat 2 things different in any way as identical? I would like to see the reasoning behind that claim, if so. Sceintific laws are true of entities different in terms of time and space - the dropping of an apple is never 100% identical with the dropping of another, they are identical to the objectively true law that dropped apples will always fall to earth.

 

[Knotted]

Hello Knotted. Can you run past me one more time, what evidence/argument you have for 1+1=2 not being objectvely true. As is demonstrated in my last post, I always misunderstand what you are saying. Or can anyone else explain what you are saying? I mean, explicitly can you show the working out, or structure of the argument? If not, then I will not say that you are right, but there is no way that I can disagree

Well first of all is there a generally agreed definition of 1+1=2? No there isn't.

Hilbert the formalist defined units as marks on paper and 2 is symbol used to represent 1+1 so in the formalist point of view 1+1=2 by definition. A bit like Pegasus has wings by definition.

The logicists (Frege, Russell & Whitehead) tried to use pure logic to define numbers. Russell and Whitehead managed to prove that 1+1=2 somewhere deep into their book called Principia Mathematica.

[At the minute I'm trying to remember who it was who defined numbers as sets, something like 0={}, 1={{}}, 2={{},{{}}},... was it Russell and Whitehead? Anyone?]

Its striking to me that these points of view do not refer to the real physical world at all. I think there is a good reason for this. If you do define numbers with reference to the real world then you end up with allsorts of different definitions using different circumstances and different interpretations. Its just not particularly fruitful.

Incidently both the formalist and the logicist strategies are flawed. They are both injered by Godel's incompleteness theorems (especially the former) and the latter uses axioms that Russell himself admitted were forced.

So all I am saying is that the truth of 1+1=2 is dependent on definition. Pick one or use your own and we'll discuss it.

 

[Knotted]

Maths could easily represent what is going on there. If maths could not function in models representing plastecine, then it would be of very limited use in science.

I can't see any reason this would be true. I keep making the analogy with impressionism. An impressionist painting does not literally represent well defined objects but it still manages to represent them. Exactness and effectiveness are not the same thing.

So did I misundersatnd the reasoning behind it in my last post? - I mean Knotted cannot be claiming anything other than fictional stauts to 1+1=1.

The statement 1+1=1 is perfectly true if you interpret it correctly. As is 1+1=2.

 

[J77]

Yes, but that speed of light analogy shows that you do not have to be able to kick something for it to be real. Imo. Do you disagree

...but you can kick the speed of light.

Well, not kick, but measure it.

eg. it takes about 1.5 seconds for light to reach us from the moon.

It's physical in that sense.

 

[118118]

Maths could easily represent what is going on there. If maths could not function in models representing plastecine, then it would be of very limited use in science

I keep making the analogy with impressionism. An impressionist painting does not literally represent well defined objects but it still manages to represent them. Exactness and effectiveness are not the same thing.

Could you explain what this has got to do with objcetive truth?

I can't see any reason this would be true

Surely your not claiming that mathematics conjoined with certain other ideas could not represent plastecine! -Please don't go off on one about sets, or effectiveness (like I say, I'm still unsure why you think this needs to be brought up all the time - I mean what role it plays in your argument), give an answer to the question, backed up with an argument if possible.

The statement 1+1=1 is perfectly true if you interpret it correctly. As is 1+1=2.

So 1+1=2 is not a fictional truth if it is interpreted correctly?

 

[118118]

So all I am saying is that the truth of 1+1=2 is dependent on definition. Pick one or use your own and we'll discuss it

I can't really as I do not know anything on the subject. Are you saying that you agree that in some definitions 1+1=2 is objetively true? Is one definition correct?

 

[118118]

...but you can kick the speed of light.

Well, not kick, but measure it.

eg. it takes about 1.5 seconds for light to reach us from the moon.

It's physical in that sense.

You could say that 1+1=2, by counting. Dunno.

 

[118118]

Anyway, I don't understand, as I thought that the argument was that 1+1=2 could not represent reality in certain situations. But 1+1=2 can still represent plastecine, but only with a little fleshing out. I don't want to argue as to whether 1+1=1 is true if interpreted properly, "the moon is made of cheese" is true if interpreted properly. If on the basis of that you went around claiming that the moon is not made of "space rock" but rather cheese, it would be rediculous.

 

[Knotted]

Could you explain what this has got to do with objcetive truth? Surely your not claiming that mathematics conjoined with certain other ideas could not represent plastecine! -Please don't go off on one about sets, or effectiveness (like I say, I'm still unsure why you think this needs to be brought up all the time - I mean what role it plays in your argument), give an answer to the question, backed up with an argument if possible.

How exact would you like your representation, sir?

So 1+1=2 is not a fictional truth if it is interpreted correctly?

What's wrong with saying that the objects of 1+1=2 (namely the "1"'s, the "+", the "=" and the "2") are fictional but if interpreted as meaning something then the whole statement is a truth.

If statement "the speed of light is constant" is true it does not mean that "the", "speed", "of", "light", "is" and "constant" are real objects.

The subject matter of science or for that matter philosophy, history, sociology etc. etc. are not words and other symbols. Why do you think the subject matter of mathematics is mathematical symbols? This seems especially absurd if you think there is an innate reality to mathematics.

 

[Jonti]

A purported definition of "number" has to correspond to that which we are trying to define, so any system which does not succeed in making 1+1=2 has to be judged unsuccessful.

At the minute I'm trying to remember who it was who defined numbers as sets, something like 0={}, 1={{}}, 2={{},{{}}},... was it Russell and Whitehead? Anyone?

Frege and Russell both proposed a set-theoretic definition of natural numbers in which each natural number n is defined as the set whose members each have n elements. But I think you may have in mind Cantor's definition. Cantor argued that we can define an empty set, a collection which contains nothing (i.e. from nothing we obtain something -- the empty set). The empty set contains no members, and we can write this as ...

emptyset = 0

There's only one such emptyset, so we can also say that the set which contains all the empty sets has just one member

{emptyset} = 1

Now, for a two element set, we need only consider emptyset and {emptyset} together

{emptyset, {emptyset}} = 2

and so on. (source -- see 2.2)

Its striking to me that these points of view do not refer to the real physical world at all. I think there is a good reason for this. If you do define numbers with reference to the real world then you end up with allsorts of different definitions using different circumstances and different interpretations. Its just not particularly fruitful.

Yes, I agree. And it seems that by not referring to the real world, these theories force us towards the conclusion that numbers arise out of the nature of thought itself (or from the nature of all possible worlds) rather than out of any particular contingent relationship with the world.

According to Wikipedia here, John von Neumann's similar definition of an ordinal number does work (meaning it survives Russell's Paradox), and the finite cardinal numbers can be defined from von Neumann's ordinals by means of the Axiom of Choice. "The axiom of Infinity then assures that the set N of all natural numbers exists. It is easy to show that the above definition satisfies the Peano axioms. It also (in contrast to some alternative definitions) has the property that each natural number n is a set with exactly n elements: {0,1,2,...,n-1}"

Seems like that's the best current definition.

 

[118118]

How exact would you like your representation, sir?

As exact as 1+1=1?

Anyway, if you accept that mathematics is objectively true, then you are led to an objective ontology if you take maths at face value, as "there is a prime number under 1000000" if objectively true, means that there there exists a prime number 1000000. Suppose you could say that whatever '1000000' refers to (may have mized things up a bit there) is not the subject matter of mathematics, but I don't see this myself.

the whole statement is a truth

An objective truth?

Sorry if I come across as annoying, I've spent alot of time on this thread.

 

[118118]

I've spend several full days on this thread, and as it turns out the person I was arguing with agreed with me from the start. What was the point in that then

 

[Jonti]

Ah, yes. One has to read Knotted *very* carefully to be able to tell when he's agreeing with one.
I've noticed that myself

 

[Knotted]

As exact as 1+1=1?

Anyway, if you accept that mathematics is objectively true, then you are led to an objective ontology if you take maths at face value, as "there is a prime number under 1000000" if objectively true, means that there there exists a prime number 1000000. Suppose you could say that whatever '1000000' refers to (may have mized things up a bit there) is not the subject matter of mathematics, but I don't see this myself.

If you take the Odyssey at face value, as "Pegasus has wings" if objectively true, means that Pegasus exists and has wings. No?

Does 1000000 have to refer to anything? Does "Pegasus" have to refer to anything? (Don't try to answer that and I won't either. )

An objective truth?

Sorry if I come across as annoying, I've spent alot of time on this thread.

The only thing that's annoying is your assumption that the topic is easy. I think you are assuming this less and less, though.

Look up Zermelo-Fraenkel set theory if you want to see how mathematicians (generally) agree on the fundamentals of grouping together seperable objects. Its a bit of a headache to the uninitiated.

 

[Knotted]

Ah, yes. One has to read Knotted *very* carefully to be able to tell when he's agreeing with one.
I've noticed that myself

I have to put my thoughts down very carefully to avoid changing my thoughts in the process. I hate it when 118118 tells me to explain my chain of reasoning, I'm much happier when reasoning isn't in chains.

Saying something which isn't either trivial or self-contradictory is very difficult on this topic in my opinion.

But I did say way back at the beginning that I was cautiously inclined to mathematical realism. I have'nt even really begun to explain this except for the need for caution.

 

[118118]

If you take the Odyssey at face value, as "Pegasus has wings" if objectively true, means that Pegasus exists and has wings. No?

Does 1000000 have to refer to anything? Does "Pegasus" have to refer to anything?

When I say "face value" I do not mean that the things which are being talked about are real. I don't consider this face value, or there would be many confused plebians But, if w2as objectively true, and not just fiction, then yes, Pegasus would exist.

If '1000000' did not refer to anything then the statement would be equivalent to "There is a prime number under hfduisho", whcih could not be true or false as the sentence is meaningless.

Please, don't let me post on this thread again. I haven't done any work in 5 days.

 

[118118]

I'm much happier when reasoning isn't in chains

How can you justify that, out of interest. Doesn't that just lead to wooly thinking?

 

[Jonti]

Saying something which isn't either trivial or self-contradictory is very difficult on this topic in my opinion.

Heh!

118118, why don't you start a thread (or post on this one); something on a topic which is relevant to the work you're supposed to be doing?

Meanwhile, I'll try to get Knotted to say why he feels a cautious mathematical realism is the right response to the question of the ontology of the natural numbers... Hey, Knotted! 'Fess up, geezer!!

That should do it

 

[118118]

118118, why don't you start a thread (or post on this one); something on a topic which is relevant to the work you're supposed to be doing?

You wouldn't understand it

Nah, I'm just supposed to be reading.

 

[Knotted]

How can you justify that, out of interest. Doesn't that just lead to wooly thinking?

Maybe, but its not a problem if you know how to use shears.

 

[Knotted]

Meanwhile, I'll try to get Knotted to say why he feels a cautious mathematical realism is the right response to the question of the ontology of the natural numbers... Hey, Knotted! 'Fess up, geezer!!

That should do it

Let's leave it for a few days. I would recommend reading what Putnam & Quine had to say on this topic. Also Hartry Field and Penelopy Maddy.

 

[118118]

Maybe, but its not a problem if you know how to use shears.

I'm sorry I don't understand. You could be making any number of claims here. We were taught that exactness is the cornerstone of good philosophy. Alot of continetal writers seem to act as if otherwise, but if their arguments don't stand up to analysis, then they are ignored. I mean, I appreciate that this is only a bulletin board, so your'e probably here to have a "laugh" but its hardly conducive to deabate.

Its absolutely the wrong position to take!

 

[Knotted]

I'm sorry I don't understand. You could be making any number of claims here. We were taught that exactness is the cornerstone of good philosophy. Alot of continetal writers seem to act as if otherwise, but if their arguments don't stand up to analysis, then they are ignored. I mean, I appreciate that this is only a bulletin board, so your'e probably here to have a "laugh" but its hardly conducive to deabate.

Its absolutely the wrong position to take!

Exactness is important for presenting a finished product. Since I don't have a finished product and presentation isn't especially important on this board, I don't think it would be appropriate to pretend that what I say has been rigorously worked out.

More to the point thought process and the presentation of the results of that process do not coincide in my opnion. This is more true in maths than it is in other subjects. You might be able to convince yourself that you think in words, and at a stretch you might be able to convince yourself that you think in terms of rigorous logical arguments expressed in words but it is somehow harder to pretend to think in terms of rigorous logical arguments expressed in abstract symbols. It might be a good exercise to pick a random mathematics paper and ask yourself how you would go about understanding it.