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Are numbers as real as rocks?

118118 said:
I might disagree that a debt cannot have measurable properties - I can count how many sheep I had and how many I have, I have thusly measured debt.

Who said you had the sheep in the first place? As Bob was saying you are talking about 0 not -5. If you have -5 sheep then this does not mean that you have lost -5 sheep, you might have started with -5 sheep. It means that if you gain 5 sheep then you end up with 0 sheep.

118118 said:
And, haven't you just tried to show that 5 positive rocks is a convention too (you could count specks of duct etc).

Well you tell me when a speck of dust is part of a rock and when it is not. You tell me when the chemical bonds have weakened sufficiently. You tell me when a speck becomes a rock. This is something you consider 'real'. You define what you are talking about then perherps I'll agree that there is such a thing as 'two rocks' rather than just 'some rock'.
 
118118 said:
it still seems like you are saying that a consistent work of fiction has the universal proprety of being consistent. Iirc your argument was something along the lines of an alien would agree that the work was consistent - yes if they agreed that it was consistnet work of fiction, but this is to beg the question, and you don't seem to think this is all there is to it.

:confused:

Perhaps I should say the fiction has not been completely described. If any additions to the fiction are consistent with the already described fiction then we can talk about properties that the additions must necessarily fulfill. These properties are not purely dependent on the consistency of the fiction but the content of the fiction and from the logical conclusions.

I don't know how that translates into philosophical jargon like '...prior to essense', but I don't see the need to resort to it in any case.
 
Knotted said:
Who said you had the sheep in the first place? As Bob was saying you are talking about 0 not -5. If you have -5 sheep then this does not mean that you have lost -5 sheep, you might have started with -5 sheep. It means that if you gain 5 sheep then you end up with 0 sheep
Well, if I didn't I would not think I now have -5 sheep. I guess that you are saying that you cannot count -5 sheep without reference to something outside immediate reflection, unlike counting 5 sheep. I disagree that this means that -5 is unmeasurable however - I cannot measure the temperature of a glass of water without resource to something outside refection.

Knotted said:
Well you tell me when a speck of dust is part of a rock and when it is not. You tell me when the chemical bonds have weakened sufficiently. You tell me when a speck becomes a rock. This is something you consider 'real'. You define what you are talking about then perherps I'll agree that there is such a thing as 'two rocks' rather than just 'some rock'.
So not only is it convention that I have two rocks, it is also convention that this is a rock. So why are numbers any less real than rocks?
 
Jonti said:
More generally, although it's an intruiging argument, it seems *too* powerful. It seems to be saying, "If the universe were quite other than it is, things that are real in our world would no longer exist, so they're not really real." Hmmm.

My argument is that:
If numbers have an a priori existence then there is no reason to find examples of them in nature.
If we find numbers in nature then we have no reason to declare that numbers are real a priori.

I'm uncomfortable with convenient coincidences. Never trust them.

I think the way to resolve it all is to understand that mathematics is not about the formality of mathematics.

Anyway I've gone into a cryptic ramble. Its a bit like a tail spin. The nature of mathematics is something both very simple and seemingly impossible to describe. Maybe we shouldn't try. Maybe we shouldn't be talking philosophy but poetry or something.

Careful with the word 'field' by the way. It has a specific meaning.
 
118118 said:
Well, if I didn't I would not think I now have -5 sheep. I guess that you are saying that you cannot count -5 sheep without reference to something outside immediate reflection, unlike counting 5 sheep. I disagree that this means that -5 is unmeasurable however - I cannot measure the temperature of a glass of water without resource to something outside refection.

Bad example with temperature. If we're talking fahrenhite or celsius, the measurement of 0 degrees is pure convention.

Also temperature is not something you count. Negative physical quantities don't come about through counting.

118118 said:
So not only is it convention that I have two rocks, it is also convention that this is a rock. So why are numbers any less real than rocks?

Very nice. But actually yes there is no such thing as 'a rock'. There is only rock. I've got no problem with talking about 'a rock' or 'two rocks' but I recognise that these things are not well defined - certainly not the 'well defined' a mathematician would recognise.

If I were to do your work for you and come up with a definition of 'two rocks' it would involve entropy. One rock can become two rocks but not the other way round. However this is a reference to the physical set up of the world not the nature of mathematics. Can you do better?
 
Knotted said:
Perhaps I should say the fiction has not been completely described. If any additions to the fiction are consistent with the already described fiction then we can talk about properties that the additions must necessarily fulfill. These properties are not purely dependent on the consistency of the fiction but the content of the fiction and from the logical conclusions
Sorry, you've confused me. You are saying that if a work of fiction is consistent and has pegasus in then pegasus must not, be able to fly and not be able to fly? And that aliens would agree with this statement, so that fictional works can have universal properties. I dunno but I'm not entirely sure that a statement of the the form if..then... can be said to have universal conclusions unless the if is always met. I mean, fictional work is not fictional, iyswim.

Knotted said:
I don't know how that translates into philosophical jargon like '...prior to essense', but I don't see the need to resort to it in any case.
Your only down with maths jargon then ;)
 
Knotted said:
Bad example with temperature. If we're talking fahrenhite or celsius, the measurement of 0 degrees is pure convention
I don't think I was using it as an example of counting negative temperature, but...
Knotted said:
Also temperature is not something you count. Negative physical quantities don't come about through counting
I was just saying that we can only meqasure some things via experience
I disagree that this means that -5 is unmeasurable however - I cannot measure the temperature of a glass of water without resource to something outside refection.
I may have misunderstood your original point and rethought it.


You tell me when a speck becomes a rock.
So not only is it convention that I have two rocks, it is also convention that this is a rock. So why are numbers any less real than rocks?
There is only rock. I've got no problem with talking about 'a rock' or 'two rocks' but I recognise that these things are not well defined - certainly not the 'well defined' a mathematician would recognise.
A speck is rock, but you cannot say how many instances of "a rock" it is - so that we could count however many instances of a rock as we liked so that 1+1 could = 1. But as you say the dieffernce here is just that the term is porrly defined - we do not know what we are counting - its not that addiotion itself has altered.

Eta: I mean, to say, that 1+1=1 - but only when the things counted are poorly definied, isn't much of a claim. If standard mathematics is only wrong when we are "wrong" about something, then so what.

Lol. I may have lost the structure of what you are arguing a little.
 
118118 said:
Sorry, you've confused me. You are saying that if a work of fiction is consistent and has pegasus in then pegasus must not, be able to fly and not be able to fly? And that aliens would agree with this statement, so that fictional works can have universal properties. I dunno but I'm not entirely sure that a statement of the the form if..then... can be said to have universal conclusions unless the if is always met.

This last is a very good point. With traditional classical logic and for that matter various minimal logics then you are right - falsehood implies anything.

I think you've uncovered a niggling insecurity I've had about what I'm saying.

However I firmly think you can meaningfully use 'if A then B' statements with A being false. That is we can successfully enquire into whether 'if A then B' is true or not. Non of our beliefs are ever quite correct, it doesn't mean our belief are useless. Also where would we be without flights of fancy, thought experiments etc. I don't fear the usefulness of myth. The dismissal of fiction is for Karl Popper and Joseph Stalin and other insecure philistines.

I like relevant logic.

118118 said:
Your only down with maths jargon then ;)

I try not to use it. If I have its been subconscious. Kick me if I do it again.
 
My argument is that:
If numbers have an a priori existence then there is no reason to find examples of them in nature.
If we find numbers in nature then we have no reason to declare that numbers are real a priori.
What does have a priori existence? Can you explain these claims? I think that you could say that numbers do not exist in nature and are not fictional.
 
118118 said:
A speck is rock, but you cannot say how many instances of "a rock" it is - so that we could count however many instances of a rock as we liked so that 1+1 could = 1. But as you say the dieffernce here is just that the term is porrly defined - we do not know what we are counting - its not that addiotion itself has altered.

Yes I agree here. What I would conclude is that our knowledge of the nature of addition is not obtained empirically.

118118 said:
Eta: I mean, to say, that 1+1=1 - but only when the things counted are poorly definied, isn't much of a claim. If standard mathematics is only wrong when we are "wrong" about something, then so what.

I think maths as a subject exists as a compromise between
A) Describing the world literally.
B) Being a useful and suitably rich tool
C) Following its own internal logic with elegance

Saying 1+1=1 follows A&C beautifully but neglects B completely.
Saying 1+1=3 isn't very good on any score but its not terrible on any of them.
Using that horrid arithmetic with different units and different additions that I described earlier scores well on A but very poorly on B&C.

Old fashioned 1+1=2 is good on all scores, but I think its far from perfect on A, and perhaps its not all its cracked up to be on B&C but it seems pretty damn good. But damn good is just damn nice, not real.

Also I think A is overated and often mistaken for B.

118118 said:
Lol. I may have lost the structure of what you are arguing a little.

Good. I'm kicking down structure.:D
 
I give in. I can never tell if you've understood my point and are arguing against, or whether you have gone off on a tangent. Oh well.
 
118118 said:
What does have a priori existence? Can you explain these claims?

For the minute could you not just be happy with the feel of what I'm saying. I don't expect to be able to pin it down exactly.

118118 said:
I think that you could say that numbers do not exist in nature and are not fictional.

This conclusion is consistent with what I'm saying, but its what I perhaps wrongly call platonic idealism which I thought you were trying to avoid.

I'm fairly sure its wrong but its the best wrong going IMO.:)
 
118118 said:
I give in. I can never tell if you've understood my point and are arguing against, or whether you have gone off on a tangent. Oh well.

I share your habit of not reading things carefully.
 
I think maths as a subject exists as a compromise between
A) Describing the world literally.
B) Being a useful and suitably rich tool
C) Following its own internal logic with elegance

Saying 1+1=1 follows A&C beautifully but neglects B completely.
The point I was making, I don't know if it is important, is that 1+1 can = 1 and decribe the world only if the things we are counting are poorly defined.

With a well defined consistent definition of what a single rock is (which you say is not prsent in your exmaple of an alternative maths), then 1+1=1 is implausible as a descrption.
 
Just to go back to the beginning again :

Rocks are rocks and are present whether humans or any other animals are present, rocks therefore are real.

Numbers, maths, is learnt by humans as they grow older so as a theory that seems to make sense but is not already present in humans numbers are a constructed theory.

If that is in dispute and it is suggested that humans, without being taught maths know that 2 rocks are 1 more than 1 rock, that would suggest that numbers are innate. If numbers are innate it is likely that other animals also understand them, the rabbit with 6 babies counts them all to determine all are back in the burrow for example.

So :

1) rocks are rocks and always were.

2) numbers may be a human developed theory that seems to make sense so it is taught to us.

3) numbers are innate and other animals also understand them perhaps in a basic way.

making any sense?
 
I live to work!

Everyone has acquitted themselves well on this thread, which usually means that the point under discussion is a dubious one. Knotted's post (75) was told to me by a friend of mine (glad I have had a while to ponder it), and although it relies on ambiguity between two people, 'if' (that word again) consensus is reached then the axiom holds. Where difference between any 2 object is perceived then the use of numbers must occur. At least to my mind.
 
I think maths as a subject exists as a compromise between
A) Describing the world literally.
B) Being a useful and suitably rich tool
C) Following its own internal logic with elegance

Saying 1+1=1 follows A&C beautifully but neglects B completely.
If these three criteria are objective, then could we not just say that there is a perfect mathematics, just that we have not found it yet. So some kind of numbers do exist independent of the mind, its just that we have some evidence that the number system we do use now is not those numbers. Not that I would agree with that, as I can't see 1+1=1 describes the world literally, as well as 1+1=2, unless the the things being counted are poorly defined. So that mathematics is only ambiguous if the thinsg being counted are.

Knotted: I thinki that the fictional concept of maths is suppsoed to be tenable. But, how do you explain that something that is only true in the story, is true in science. 1+1=2 is not only true in the story of mathematics, but also in scientific work. If you claim that it is not true that 1+1=2 in science, then my encyclopedia disagrees with you. And, I would add that 1+1=3 cannot be true in scientific study, as that would be incompatible with what is.

Eta: What I quoted above is also true of phillsophy. Does that mean that no philosophy is a fiction :confused:
 
118118 said:
The point I was making, I don't know if it is important, is that 1+1 can = 1 and decribe the world only if the things we are counting are poorly defined.

With a well defined consistent definition of what a single rock is (which you say is not prsent in your exmaple of an alternative maths), then 1+1=1 is implausible as a descrption.

Not sure about that, but it misses the point. I don't think arithmetic is a direct representation of reality and more than that I don't think it would be desirable either.
 
weltweit said:
Just to go back to the beginning again :

Rocks are rocks and are present whether humans or any other animals are present, rocks therefore are real.

Numbers, maths, is learnt by humans as they grow older so as a theory that seems to make sense but is not already present in humans numbers are a constructed theory.

If that is in dispute and it is suggested that humans, without being taught maths know that 2 rocks are 1 more than 1 rock, that would suggest that numbers are innate. If numbers are innate it is likely that other animals also understand them, the rabbit with 6 babies counts them all to determine all are back in the burrow for example.

So :

1) rocks are rocks and always were.

2) numbers may be a human developed theory that seems to make sense so it is taught to us.

3) numbers are innate and other animals also understand them perhaps in a basic way.

making any sense?

I suppose that we need three things before we can have a notion of arithmetic.
Firstly we need to be able to seperate things from each other. I don't think this requires much more than good eyesight.
Secondly we need to be able to group like with like. I understand this is quite a problem in artificial intelligence but most if not all animals seem quite competent as far as I know.
Thirdly we need to be able to have a concept of abstract representation and a notion of equality of these representations. I would think this is a nearly uniquely human.
 
118118 said:
If these three criteria are objective, then could we not just say that there is a perfect mathematics, just that we have not found it yet.

I was thinking in terms of mathematics as an educational/scientific institution and the subtle tensions within it.

I don't think there is a perfect anything. There is always imperfection, conflict and compromise.

118118 said:
So some kind of numbers do exist independent of the mind, its just that we have some evidence that the number system we do use now is not those numbers. Not that I would agree with that, as I can't see 1+1=1 describes the world literally, as well as 1+1=2, unless the the things being counted are poorly defined. So that mathematics is only ambiguous if the thinsg being counted are.

I'm still going to say that direct representation of the world is unimportant - except for educational purposes. I think 1+1=1 always works whereas 1+1=2 only usually works. The moral is not that the former is superior but that neither statement is a literal statement about the physical world.

118118 said:
Knotted: I thinki that the fictional concept of maths is suppsoed to be tenable. But, how do you explain that something that is only true in the story, is true in science. 1+1=2 is not only true in the story of mathematics, but also in scientific work. If you claim that it is not true that 1+1=2 in science, then my encyclopedia disagrees with you. And, I would add that 1+1=3 cannot be true in scientific study, as that would be incompatible with what is.

Hartry Field wrote a paper describing Newtonian mechanics without reference to arithmetic. I haven't read this myself, but there seems to be a consensus that it has demolished the view that arithmetic is essential to understand science.

Plus as I have shown very simply that an arithmetic with 1+1=3 is just as good a tool for scientific study.

Maybe number and standard arithmetic always creeps in disguise, and I tend to believe it does, but it needn't be a starting point and can be represented with all sorts of formalities with all sorts of physical interpretations.

So the idea that the formalities of mathematics are real is a nonsense. If there is a reality to mathematics we do not have the language to even refer to it. Outrageous as it sounds, this is pretty much what I feel. I see the formality of mathematics as being like an artist's impression of the underlying 'mathematical reality'. Just as the impressionists showed that art does not need to be literal in order to depict its subject well, I don't think mathematics need be literal in order to describe reality well.

118118 said:
Eta: What I quoted above is also true of phillsophy. Does that mean that no philosophy is a fiction :confused:

No, but I'd be hard pressed to prove philosophy is not just fiction in any case.
 
Knotted said:
Not sure about that, but it misses the point. I don't think arithmetic is a direct representation of reality and more than that I don't think it would be desirable either.
I don't know what you mean by direct representation of reality. But you have included fits reality or something similar as a description of a good arithmetic. And I don't think that any other arithmetic than the one that we had would do so as well :confused: are we talking past each other?
 
Plus as I have shown very simply that an arithmetic with 1+1=3 is just as good a tool for scientific study
I have read this post several times, and I see nothing in it. Can you imagine atriangle without three sides - the idea is just as rediculous.
 
118118 said:
I don't know what you mean by direct representation of reality. But you have included fits reality or something similar as a description of a good arithmetic. And I don't think that any other arithmetic than the one that we had would do so as well :confused: are we talking past each other?

With a bit of thought I've decided that the main reason that arithmetic should directly represent reality is educational. I think it is a bad educational strategy to introduce numbers as abstract entities following rules which ultimately form the bedrock of good scientific theories to small children. I think we're best off with one apple plus one apple makes two apples.
 
Its called sceptcism knotted. No offense, but unless you can provide references of a professor that holds your view, I am inclined ti think that you are talking s*. By saying tat we could have an alyernative mathematics to describe how plastecine moulds together, is to say that ,mathematics describes how stars collapse. And you deny this. So you have made a fundamental mistake from the outset.
 
Knotted said:
With a bit of thought I've decided that the main reason that arithmetic should directly represent reality is educational
So any and all mathemtics is the same - just true because of education. - so how can it be truie in science?
 
118118 said:
I have read this post several times, and I see nothing in it. Can you imagine atriangle without three sides - the idea is just as rediculous.

But triangles don't exist either. It shouldn't be surprising if mathematicians use a geometry that is consistent with their arithmetic. That doesn't say anything about the absolute nature of either.
 
I'mnot sure that you've answered any of my questions beyond reststing that it doesn't matter because you don't think so. This argument is gettinh noweher, its not just my imagination?
 
Knotted said:
But triangles don't exist either. It shouldn't be surprising if mathematicians use a geometry that is consistent with their arithmetic. That doesn't say anything about the absolute nature of either.
Ok, I accept that if triangular things do not exist then traingles cannot exist. Do triangular things exist? The world is just a unstructuredmess, and we impose meaning on it - I thought that was what I was arguing for :confused: (a bit drunk, sorry)
 
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