Are numbers as real as rocks?
- Thread starter Jonti
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weltweit said:
In my view animals have their own concepts.
To begin with they don't see the world through the same lenses as the human eye. Many animals see much more then we do, some have almost no sight, others are blind. Each of them must have a specific system of values and reasoning which -to begin with - they need to survive.
salaam.
Bob_the_lost
Elsewhere
Complex numbers make sense, they are vital in fact for a lot of mathematical concepts, why are they any less god given again?Knotted said:
Bob_the_lost said:Complex numbers make sense, they are vital in fact for a lot of mathematical concepts, why are they any less god given again?
Well instead of complex numbers you could talk about the direct product of the field of reals with itself with the quotient generated by (0,a)x(0,a)=(-a^2,0) factored out. You do not need a notion of complex numbers to talk about complex numbers. I don't think this is true of natural numbers and even when you try to describe them axiomatically there is always something about the axioms (or the formalism of the axioms) which has an implicit notion of number - in my opinion at least.
Having said that, real numbers seem to describe space very well whether or not you regard them as pure formalism and complex numbers have a crucial role in quantum mechanics. The latter example is particularly strange as complex numbers have existed as a mathematical curiosity long before Planck, Bohr and Schrodinger. Obtuse, abstract notions of mathematical completeness - for example being able to solve the equation x^2+1=0 - end up having real scientific meaning. The empiricist outlook seems very weak in this light.
Bob_the_lost
Elsewhere
So to avoid using these number you can use other means of representation...Knotted said:
We'll ignore the impracticality of trying to use that method for some of the really fun stuff you can do with complex numbers
Which makes them just as real as integer numbers again? Either they all are or none are. I prefer the latter.
What a cool thread! I think that just because you cant see, hear or touch something doesn't mean its not "real". For instance you can't see, hear or touch "politics"... or an emotion, such as "anger" but, we are all in agreement that such things exist. It's true that such concepts depend upon strong consensus, right out into the full seven dgrees of separation of our civilisations, and the accumlated body academic work that is "mathematics" but when such consesus about the about a concept like numbers is so overwhelming, we can do little award them their own place in our reality. Even if the nature of that reality itself is ultimately subjective. Imagine another reality where Pi has a different value and where salt and vinegar crisps taste totally different! 
Numbers and maths are interesting in that they form a self-referential system. Most of our systems of understanding and interpretation are based on external experience and provable only by an assumption that our external experience in some sense reflects 'reality', whereas numbers prove themselves regardless of the delusions of existence. The manifestation of mathematical phenomena in our reality experience is not the most fundamental proof of their solidity, but rather the fact that no external reality is necessary for them to continue to make sense makes them fascinating.
Here's some thinking on this.
Can fiction have objective, universal properties? I think it can if it is consistent. Miss Scarlet could not have carried out the murder in both the conservatory and the kitchen. I think that is an objective truth that an alien race would have to agree with if they could understand the story. That doesn't mean Miss Scarlet, the conervatory or the kitchen in question actually exist.
Human beings are using fictional abstractions all the time. Its perhaps a uniquely human trait and if it isn't there are very few species that share it (probably just the great apes) and non to the extent we do.
For example I might 'put myself in someone else's shoes'. I might say, 'if I were you I would accept that job offer.' That's nonsense of course. I am not you and even if I were I would do what you do because I would be you and not me. Its still a very useful way of thinking though and I think because of that humans are prone to believing that abstractions are real.
Keep the above in the back of your mind while I move to a different topic.
What is the subject matter of mathematics? Is it about mathematical objects in themselves? I don't think so. I think the meat of the subject is the proofs about mathematical objects.
For example take Fermat's last theorem. This was stated by Fermat centuries ago and many brilliant mathematicians have tried but failed to prove. Recently Andrew Wiles came up with a difficult proof relying on some very sophisticated modern mathematics. However the point I want to make is that nobody seriously doubted the truth of Fermat's last theorem in the first place and furthermore the truth of it is without consequence even for pure mathematicians. However the proof of the theorem, and more importantly the failed attempts of the proof, have involved major discoveries of mathematical techniques.
So my thinking is that mathematical realism has the right flavour - but not because numbers are real, but because there are objective, unobvious and surprising useful mathematical properties but these only become properly apparent when a mathematical proof is discovered. Proof puts the theorem/conjecture into a broader methodical context.
Can fiction have objective, universal properties? I think it can if it is consistent. Miss Scarlet could not have carried out the murder in both the conservatory and the kitchen. I think that is an objective truth that an alien race would have to agree with if they could understand the story. That doesn't mean Miss Scarlet, the conervatory or the kitchen in question actually exist.
Human beings are using fictional abstractions all the time. Its perhaps a uniquely human trait and if it isn't there are very few species that share it (probably just the great apes) and non to the extent we do.
For example I might 'put myself in someone else's shoes'. I might say, 'if I were you I would accept that job offer.' That's nonsense of course. I am not you and even if I were I would do what you do because I would be you and not me. Its still a very useful way of thinking though and I think because of that humans are prone to believing that abstractions are real.
Keep the above in the back of your mind while I move to a different topic.
What is the subject matter of mathematics? Is it about mathematical objects in themselves? I don't think so. I think the meat of the subject is the proofs about mathematical objects.
For example take Fermat's last theorem. This was stated by Fermat centuries ago and many brilliant mathematicians have tried but failed to prove. Recently Andrew Wiles came up with a difficult proof relying on some very sophisticated modern mathematics. However the point I want to make is that nobody seriously doubted the truth of Fermat's last theorem in the first place and furthermore the truth of it is without consequence even for pure mathematicians. However the proof of the theorem, and more importantly the failed attempts of the proof, have involved major discoveries of mathematical techniques.
So my thinking is that mathematical realism has the right flavour - but not because numbers are real, but because there are objective, unobvious and surprising useful mathematical properties but these only become properly apparent when a mathematical proof is discovered. Proof puts the theorem/conjecture into a broader methodical context.
I would have thought that we have evidence thast numbers are real - arithmetic I would have thought feature in natural laws, unlike unicorns that can only exist possibly (they are fiction)Knotted said:
I suppose you could say that unicorns feature in our thought experiments, but it isn't very intuitive that what does has the same status as what does in physics experiments.
Unicorns are not as real as numbers.
If a fiction has the universal property of consistency, then it has a universal property - consistency.
But you have not really argued for consistency being a universal property.
Eta: Maybe you have.
Presuambely to understand the story they must agree that pegasus can fly.Knotted said:
So, you may as well just say: if they agree that pegsus can fly then there is a universal property to pegasus: flying.
But this says next to nothing, what reason do you have to believe that they would agree that pegasus can fly?
Again, you have not really argued for the belief that fictional entities are universal. Just that if it were, they would be. I think, anyway.
Jonti
what the dormouse said
Yes, that's what it's all about alright.
What makes the objects seem real (not saying they're not) is the fact that putative properties are apparent. One then tries to prove these, and it sometimes turns out to be intruigingly hard; or maybe impossible; or perhaps that one's intuition was plumb wrong. That's how mathematical understanding is advanced.
But numbers in particular aren't just "theories about numbers" are they? They're applied math as well
Alright I'll do a long winded version.
Its not necessarily obvious what properties a fiction does or does not have. However if it is consistent it certainly forces the fiction to not have contradictory properties and to say that the fiction does not have such and such a property is to give describe a property of the fiction.
Of course you could say that this sort of property is just a consequence of consistency and so it is not really a proper property at all. However I don't think this is particularly sensible suggestion if the consequences of consistency are unobvious. It makes you look foolish if you say something is trivial but you have no idea why it is true.
Well, I did not mention pegasus. However it is not a question of agreeing that pegasus can fly, but agreeing that pegasus can fly within the fiction. So I am not arguing that fictional entities are universal, but that certain statements with reference to certain types of fiction are universal. Nobody need agree on the truth of statements within the fiction but an honest observers are likely to be forced to agree on the contents of the fiction and even the logical consequences of the contents of the fiction.
Surely you would agree that in Homer's Odyssey, Pegasus could fly. You would also surely agree that in Homer's Odyssey, Pegasus could not both fly and not fly at the same time.
I think the second truth is objective and universal (even if it is trivial) whatever we may think about the Odyssey and Pegasus. So my conclusion is that the discovery of objective, universal truths does not mean that the subject matter of the truth is objective or universal.
Where's the smiley for 'heurrrrgh'?
118118 said:
Its not necessarily obvious what properties a fiction does or does not have. However if it is consistent it certainly forces the fiction to not have contradictory properties and to say that the fiction does not have such and such a property is to give describe a property of the fiction.
Of course you could say that this sort of property is just a consequence of consistency and so it is not really a proper property at all. However I don't think this is particularly sensible suggestion if the consequences of consistency are unobvious. It makes you look foolish if you say something is trivial but you have no idea why it is true.
118118 said:
Well, I did not mention pegasus. However it is not a question of agreeing that pegasus can fly, but agreeing that pegasus can fly within the fiction. So I am not arguing that fictional entities are universal, but that certain statements with reference to certain types of fiction are universal. Nobody need agree on the truth of statements within the fiction but an honest observers are likely to be forced to agree on the contents of the fiction and even the logical consequences of the contents of the fiction.
Surely you would agree that in Homer's Odyssey, Pegasus could fly. You would also surely agree that in Homer's Odyssey, Pegasus could not both fly and not fly at the same time.
I think the second truth is objective and universal (even if it is trivial) whatever we may think about the Odyssey and Pegasus. So my conclusion is that the discovery of objective, universal truths does not mean that the subject matter of the truth is objective or universal.
Where's the smiley for 'heurrrrgh'?
Jonti said:Yes, that's what it's all about alright.
What makes the objects seem real (not saying they're not) is the fact that putative properties are apparent. One then tries to prove these, and it sometimes turns out to be intruigingly hard; or maybe impossible; or perhaps that one's intuition was plumb wrong. That's how mathematical understanding is advanced.
But numbers in particular aren't just "theories about numbers" are they? They're applied math as well
But surely what makes something actually real as opposed to seem real is if its existence has a consequence. Its arguable that number properties have a consequence in how the world is structured, but I don't think you can argue that numbers themselves have consequences.
Besides numbers seem to be derived from natural common sensical human behaviour. You group things together, you abitrarily decide what can be in that group and you arbitrarily decide that things in that group can't merge or cause each other to divide. In short the whole number business just doesn't stand up to rigorous scrutiny (or pedantry if you prefer) so it doesn't seem to be a very good basis for an utterly precise, rigorous subject. Why should something so beautiful as the number six be based upon the mucky business of collecting miserable physical objects?
I can't help believe that number properties have an objective existence even though numbers themselves don't. I really think that I'm in deep philosophical water here. Disembodied properties doesn't sound good. Somebody should be able to shoot me down with a well aimed reference from maybe Quine or Rorty or someone? Give it your best shot.
Jonti
what the dormouse said
Things and numbers are clearly not the same sort of thing, and this neatly nails down a crucial distinction. Numbers are still real, for their properties are the same to everyone.Knotted said:
So yes, that said, it's a principle of scientific reason that it makes no sense to talk about existant things that have no effect or consequences at all. If there's no way, in principle, ever, to discover any effects of a thing's existence, well then that thing might as well not exist. Fair enough.
But I wouldn't agree that number is based on arbitrarily deciding that things in some group can't merge or cause each other to divide. Even a fluid world of endless merging and dividing implies number. To divide is make a distinction. To merge to is to remove one, to make two into one.
It's marvellous, isn't it, how every number is different. Seventeen is one of my favourites.
Knotted, I'm going to disagree with you. A work of fiction can be inconsistent. And its consistency is a higher order "property" (? maybe I have this the wrong way around again). You seem to be making some kind of claim like: coinsistency preceeds essence. Which is doesn't, because one has only created a consistent work of fiction once one have created a work of fiction that has events, characters, a plot etc.
Or maybe you are making a claim like: a good work of fiction is always consistent, goodness preceeds essence, so consistency preceeds "essence".
Also, I think that there only being 1 big bang would be very different to there being 2. Instances of number are no less number.
Oh, I see, (-5) pigs. In a way you can get hold of -5 pigs, you can remove 5 pigs - thus 5 pigs have been taken away by you. Taking away 5 pigs = (-5) pigs. You have (-5) pigs. Etc.
And you can certainly count (-5) pigs. 10-5=5, I can count that.
I can see how (-5) pigs isn't quite as visceral as 5 pigs, but many people would argue that you can see absence as well as presence.
Yeah, I disagree with this. Its just a statement really, with not much attempt to justify it, so I'm not sure I can say much more. Its not a very good argument, but I find logical laws (see other thread) too immutable to be invented. I also think that its likely that to think that we invented the number 1, but there was 1 big bang, unlikely - as you would need some kind of outlandish explanation for how our invented concepts were instantaited prior to them being invented.Bob_the_lost said:
Ontology: Theory of what exists
Instantiated: To make concrete? Something like that anyway
An insatnce of.Absence: Something which is not there is, absent
Presence: Something which is there is, present
I mean ffs bob you have not explained anything. Get a grip. Is 1+1=2 not independent of the mind (if it is as you say invented)? Could a group all agree that 1+1=3, and be just as correct as another group that holds that 1+1=2, so that that 1+1=both 2 and 3? Mathematicians tend to agree with one another on what 1+1=, how can you explain that?
118118 said:
I'm not sure about that. Maybe we could talk about a framework for a fiction and we could perhaps talk about 'local consistency' - this well defined subsection of the fiction is consistent. In that case we could talk about consistency preceeding essence. But really I'm not so much concerned with the question of higher order properties and essenses and the like. I'm just describing how mathematicians see their subject. They may or may not see it as fictional, but they would certainly see it as necessarily consistent. Reductio ad absurdum plays a central role in finite mathematics at least.
118118 said:
No.
118118 said:
I disagree. It depends on what you are counting and in what type of world. I could happily define numbers to be quite different, though I would still insist that 'our' notion of number has universal properties that aliens from a different universe would have to agree on if they are honest and can understand it. I might demonstrate this for you (and Jonti) later...
Bob_the_lost
Elsewhere
No, that's not -5 at all. FFS did you not read my post?118118 said:
As for seeing absence that's really, really badly thought through. You could recognise that you only have 3 pigs left instead of eight but it's impossible to see -5 pigs. Again it'd be a human invention / thought process. I can argue that i could see the result of santa claus therefore claim he exists, but i'd be talking shite. The presence of presents does not mean the concept of santa is real.
Mathematics is a tool invented to help predict and model the universe. It's an amazing coincidence that it works so well. However this does not make it real, it just makes it a good model. When it's found that mathematics does not explain things then it's altered so that it does. So it's reality changes.
Read up on Newtonian physics for an example of how theory that works is not always true.
Bob_the_lost
Elsewhere
Dear gods, they all share the same delusion and therefore they are all right?118118 said:
Are you on leave from the special school of logical arguement?
Jonti
what the dormouse said
I certainly agree with this. To me, it seems so obvious as to be beyond dispute. As I put it to Alde, aliens would find the same set of primes that we find, even if they saw the universe very differently from ourselves.Knotted said:
I'm not sure how one could *prove* it though
Jonti
what the dormouse said
The gods won't help you bob. They invented the integers in the first place
How would the world be, if the properties of the integers were not ("coincidentally" !) congruent with the way things actually are?
Yeah, amazing, innit? But to write it off as just a coincidence seems a little ... weak.Bob_the_lost said:
How would the world be, if the properties of the integers were not ("coincidentally" !) congruent with the way things actually are?
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