Are numbers as real as rocks?

[fractionMan]

1+1=1 is just untrue.

I was reading about this kind of stuff and Gödel's incompleteness theorem is quite an interesting read for maths novices such as myself. I only understood about half of it mind

I think it's saying that 1+1 doesn't have to equal 2. Or that you can't prove it. Or something.

The wikipedia article on http://en.wikipedia.org/wiki/Philosophy_of_mathematics makes moree sense.

 

[118118]

I don't know, but I think that it says that 1+1=2 can not describe all mathematical truths. I vaguely remeber trying to think about whether or not it could in A-level maths: a biit of a headache iirc.

I don't think that its particularly relevent to realism or even certainty.

Besides which I don't know if I think that 1+1=2 is that certain, I just think that anyone who wnats to argue that 1+1=0 is insane.

Maybe, this is similar to the idea that I am hallucinating this conversation, yet anyone who wants to convince me that I am actually an alien floating in a bubble over the Caspian sea, would be insane.

I would be happier if people were arguing that it is not 100% undoubtable that that 1+1=2, not that 1+1=1. There's a bit of a difference, maybe.

 

[Knotted]

I'm not sure what remains of your point when it is insisted that enumeration is mathematics.

I definitely didn't say that. I would clocked myself one.

Therefore, as 0+0=1 is not mathematics, 0+0=1 is not enumeration either.

What does this refer to?

There is nothing surprising about the fact that different mathematical truths apply to 0 as to 1, so it should not be too surprising that enumerating 0 is different to eunemerating 1.

Well the point is that you cannot enumerate 0's. One apple and one apple makes two apples. One six and one six make two sixes. One zero and one zero makes one zero or three zeros or 15356 zeros.

If zeros are objects then why can you not count them?

1+1=1 is just untrue.

1+1=1 is just pixels on a computer screen. If it doesn't refer to anything it can neither be true nor false.

 

[Knotted]

I think it's saying that 1+1 doesn't have to equal 2. Or that you can't prove it. Or something.

No it says that given any consistent axiomatic system that describes natural numbers there are theorems that can be stated in that axiomatic system which are unprovable.

I'll try to dejargon that.

Consistent means you cannot say one thing and then say the opposite.

An axiom strictly speaking means 'self evident truths', although more often than not the self evidence of the truths is questionable but we can say that we agree with them for the sake of argument at least.

An axiomatic system is a system built by assuming 'axioms' and then looking at the logical consequences of these axioms.

Natural numbers are whole numbers bigger than 0 (usually including 0)

A theorem is a statement which has been proved.

In other words no single axiomatic system is completely up to the job for describing arithmetic.

 

[Knotted]

I don't think that its particularly relevent to realism or even certainty.

Godel certainly thought it was.

Edit to add:
Roughly it shows that arithmetic is more than the (finitely stated) formality of arithmetic. If I remember rightly, in Godel's thinking this shows that there is a reality to mathematics which is beyond the mere shifting of symbols. I'm inclined to cautiously agree.

Besides which I don't know if I think that 1+1=2 is that certain, I just think that anyone who wnats to argue that 1+1=0 is insane.

In a Boolean ring 1+1=0. '1' represents a statement which is true, '0' represents a statement which is false and '+' represents exclusive OR.

Maybe, this is similar to the idea that I am hallucinating this conversation, yet anyone who wants to convince me that I am actually an alien floating in a bubble over the Caspian sea, would be insane.

I would be happier if people were arguing that it is not 100% undoubtable that that 1+1=2, not that 1+1=1. There's a bit of a difference, maybe.

The reason I think that 1+1=2 is that mathematicians have agreed on it. It does the job. If I say that 1+1=0 or 1+1=1 then I have other jobs in mind. And as far as the formalities go that's the end of the story. If you want to think about why it does the job and what that job is and what other jobs there could be then we can move onto much more interesting questions.

 

[Knotted]

Wikipedia is remarkably good these days.
x+x=0 for all x in a Boolean ring
See:
http://en.wikipedia.org/wiki/Boolean_ring

 

[118118]

Well, I was saying that enumeration is part of mathematics. Counting, isn't it? Iiirc Husserl seemed to think so, he may not have.

I don't understand the "does the job" discussion. Yes it does the job, so does thsi mean that we only think its true because it does the job? Those 2 pages of philosophy of maths that I have read just state that 2+2=4 is more certain than most ideas.

Could you provide a refernce to someone who has similar views to your own?

I can't see that it is relevent that: when 1 = and 0 represent different objects (fictional or not) to normal, 1+1=0. Could you explain?

I don't find it surprising that you can't enumerate 0's in the same way as 1's, anymore than I find it surprising that 5x9=45, and 6x9=54. Its just arithmetic spewing out differnt results.

 

[118118]

If zeros are objects then why can you not count them?

See above

1+1=1 is just pixels on a computer screen. If it doesn't refer to anything it can neither be true nor false

Then how can you bne a realist? Godel would disagre

 

[118118]

Maybe its best I leave this thread alone and let you talk about Godel.

Btw, I don't think that I could count the pebbles on a beach. Doesn't mean that pebbles aren't objects.

 

[Knotted]

I don't understand the "does the job" discussion. Yes it does the job, so does thsi mean that we only think its true because it does the job? Those 2 pages of philosophy of maths that I have read just state that 2+2=4 is more certain than most ideas.

Could you provide a refernce to someone who has similar views to your own?

Well here's a good place to start:
http://www.lps.uci.edu/home/fac-staff/faculty/maddy/final p_maddy_mathematical_existence.pdf

I think that:

1) Mathematics has generated plenty of philosophical arguments but has always developed regardless of these arguments. I'm supportive of Maddy's naturalism. The role of philosophy should not be to dictate some sort of metaphysic.

2) Mathematics does not derive from science nor vice versa although there is a symbiotic relation. The considerations of axiomatic systems are driven by the need for consistency, the mathematical interest they generate and the brevity with which that mathematical interst can be expressed.

This makes me very sympathetic to much in the quoted article. The notion of 'thin realism' is vague enough to be about right.

I can't see that it is relevent that: when 1 = and 0 represent different objects (fictional or not) to normal, 1+1=0. Could you explain?

I thought I did. Check the wikipedia link. In abstract algebra '0' usually represent the additive identity and '1' usually represents the multiplicative identity. So we have the rules:
0+x=x+0=x
1*x=x*1=x
There is no reason that 1+1=0 can not be true.

I don't find it surprising that you can't enumerate 0's in the same way as 1's, anymore than I find it surprising that 5x9=45, and 6x9=54. Its just arithmetic spewing out differnt results.

So you don't find it surprising that 1+1=1 some of the time but 1+1=2 some of the time because we could be talking about one 0 and one 0 in the first instance and one 1 and one 1 in the second? It seems to outrage you at other times!

By the way I am not claiming that 0's enumerate in a different way to 1's. I'm claiming that 0's can't be enumerated full stop. You can't count them. How many 0 sheep do you see before you?

 

[Knotted]

See above
Then how can you bne a realist? Godel would disagre

I think Godel would see the symbols as representatives of mathematical objects, rather than the objects themselves.

 

[muser]

The reason I think that 1+1=2 is that mathematicians have agreed on it. It does the job. If I say that 1+1=0 or 1+1=1 then I have other jobs in mind. And as far as the formalities go that's the end of the story. If you want to think about why it does the job and what that job is and what other jobs there could be then we can move onto much more interesting questions.

I was hoping the conversation would shift into the realms of alternative mathematics. I'm not aware of any branch of maths that allows for 1 + 1 = 1, but no matter how illogical it must appear mathematician must have been tempted to create a whole new set of conditions regarding existing operator and\or creating new one. How would it affect geometry, algebra, trigonometry if you were to change the basic premise of logical mathematics.

 

[muser]

By the way I am not claiming that 0's enumerate in a different way to 1's. I'm claiming that 0's can't be enumerated full stop. You can't count them. How many 0 sheep do you see before you?

You've just assigned your object (e.g. 0) a sheep. Why?

 

[118118]

I'm not claiming that the marks on a piece of paper have any immaterial existence. I mean, I did not think we were dicussing the reality of the marks on a piece of paper.

Isn't it simply the case that your example of 1+1=0 culd be reffering to different objects than 1+1=2. That seems very likely to me.

And I want to say again that I do think that enumeration is a form of mathematics - so its not surprising that you enumerate 0's differently/not at all. I will read those links in a bit.

 

[Knotted]

You've just assigned your object (e.g. 0) a sheep. Why?

Purely in order to make myself understood.

 

[Knotted]

I'm not claiming that the marks on a piece of paper have any immaterial existence. I mean, I did not think we were dicussing the reality of the marks on a piece of paper.

Well what is a mathematical object in your view? What are mathematical truths about?

The way I see it you can argue a strong mathematical realism in two ways:
1) As an extension to scientific realism.
2) As a form of platonism.

I'm at a loss as to understand what point of view you take.

Isn't it simply the case that your example of 1+1=0 culd be reffering to different objects than 1+1=2. That seems very likely to me.

Well yes! The former could refer to a point on a circle with a circumference of two metres. Travel two metres and you end up at the start, so 2=0.

Our notion of number can refer to many things. It could be a form of accounting, it could be to do with harmonics. I don't think there is a single physic or metaphysic of number. The number 6 could refer to six objects or it could refer to the difference between a half and a third. I don't think there is a single physic or metaphysic of number.

And I want to say again that I do think that enumeration is a form of mathematics - so its not surprising that you enumerate 0's differently/not at all. I will read those links in a bit.

I must say this point of view surprises me. I didn't expect you to say that! So why do you think 1+1=2? By the way don't worry yourself too much with the Boolean rings. Its algebraicisation (to coin a horrible word) of logic. But rings (mathematical structures that have nothing to do with a misleading name) in general are generalisations of ordinary arithmetic, some are quite exotic.

 

[muser]

Purely in order to make myself understood.

I thought that was 118 118 who posted that in response to your post

sorry, just saw that it was you. If it had been 118118, I would have asked for an explanation.

 

[118118]

So did I for a bit! Weird.

 

[Knotted]

I was hoping the conversation would shift into the realms of alternative mathematics. I'm not aware of any branch of maths that allows for 1 + 1 = 1, but no matter how illogical it must appear mathematician must have been tempted to create a whole new set of conditions regarding existing operator and\or creating new one. How would it affect geometry, algebra, trigonometry if you were to change the basic premise of logical mathematics.

I think that there are two possibilities. Either you would end up describing our usual notion of number in other terms, so that geometry, algebra etc would be essentially the same but more convoluted. Or you would be able to describe certain mathematics in terms of this new number system, but not all of it.

If you look at abstract algebra - ring theory, group theory etc. you will discover that there are lots of different operators which apply to lots of different types of object, some of which are addition like, some multiplication like, some quite alien.

As an easy example, if you imagine a number line, but instead of it going on and on for ever it is rapped up in a circle, so that when you reach 1 you are back to 0. Then 1+1=0+0=0=1, so 1+1=1. Its a reasonable, consistent alternative to arithmetic, although I suspect it is weaker in the sense that you cannot use it as broadly as ordinary arithmetic.

 

[118118]

Like I say, I apologise for getting caught in a discussion that I know very little about. One would have thought that a degree in philosophy would have taught me something!

I have only read a secondary text on the Husserl's philosophy of arithmetic, but he abondoned it so I wasn't looking for all the answers. But that is where most of my feelings on the subject come from.

Our notion of number can refer to many things. It could be a form of accounting, it could be to do with harmonics. I don't think there is a single physic or metaphysic of number. The number 6 could refer to six objects or it could refer to the difference between a half and a third. I don't think there is a single physic or metaphysic of number

The important thng seems to be to me, that when 1+1=0, you are just referring to different entities (be they fictional, objective, ideal...) you are not reffring to the objects that when added together make 2, so 1+1=0 in thiese alternatives make 1+1=2 no less true or certain.

Its just irrelevent, if your 1+1=0 could just as easily be treated as A f A g B. Where is the overlap with 1+1=2 except in the symbols being used, used which are arbitary (as the 1's e.g. are being used toi refer to different things). If both signified and signifying are different, how could it be relevent.

And I think that enumeration is mathematics, becasue what more is there to counting to 2 than adding to 1's together.

 

[Knotted]

Like I say, I apologise for getting caught in a discussion that I know very little about. One would have thought that a degree in philosophy would have taught me something!

Really don't worry about it. I think that philosophy departments generally don't encourage students to study the philosophy of maths, its just too hard. My girlfriend did a course in it as an undergraduate and the only others taking the course were post-graduates.

I've got no academic background in philosophy at all, although I done postgraduate mathematical research, so I have a good idea of what maths tastes like. That's not say I can speak with any great authority on the philosophy of maths but I do know how mathematicians operate. Hardly any are concerned about ontological issues. If you say that the whole thing is a fiction or if you say that mathematical objects are real objects, most mathematicians would just shrug and say, "at least we can agree on what we are doing." The Penelope Maddy reference I gave you makes nice comforting reading for me, "thin realism" sounds about right - we'll talk about it as if it were real whatever that means.

I have only read a secondary text on the Husserl's philosophy of arithmetic, but he abondoned it so I wasn't looking for all the answers. But that is where most of my feelings on the subject come from.

There is probably a moral to the story of Husserl abandoning the philosophy of arithmetic...

The important thng seems to be to me, that when 1+1=0, you are just referring to different entities (be they fictional, objective, ideal...) you are not reffring to the objects that when added together make 2, so 1+1=0 in thiese alternatives make 1+1=2 no less true or certain.

Absolutely. But where does the 'truth' that 1+1=2 come from?

Its just irrelevent, if your 1+1=0 could just as easily be treated as A f A g B. Where is the overlap with 1+1=2 except in the symbols being used, used which are arbitary (as the 1's e.g. are being used toi refer to different things). If both signified and signifying are different, how could it be relevent.

I don't think it is irrelevant, it aught to focus our attention on what we are talking about. When we talk about number what is that we are refering to? I can't go any further with you until you give some sort of answer to that question. This is why I offer you variety of different but similar systems. If you had a reference point you could say, "well we know that this is not true of numbers because..."

And I think that enumeration is mathematics, becasue what more is there to counting to 2 than adding to 1's together.

Well to be flippant 2 could be 3 with 1 subtracted. But what I really think you are saying is something along the lines that 1 is a more fundamental idea than 2 and we can talk about 2 in terms of 1. This is a very common point of view, including and especially amongst those who wish to describe arithmetic in axiomatic terms. But these reductionist schemes seem destined to fail. Why not just let 2 be 2.

Here's a point of view from Luitzen Brauer where 2 seems to be fundamental:

"Mathematics arises when the subject of two-ness, which results from the passage of time, is abstracted from all special occurrences. The remaining empty form [the relation of n to n+1] of the common content of all these two-nesses becomes the original intuition of mathematics and repeated unlimitedly creates new mathematical subjects." (quoted in Kline 1972, pp. 1199-2000)

 

[118118]

I don't think it is irrelevant, it aught to focus our attention on what we are talking about

Doesn't that answer the question as to whether numbers are real though? To state what the subject matter of mathehatics is, is to say whether they are eternal or not, mind independent or not, empirical or not. I can't raelly say what I believe the subject of maths is though. But I don't feel like giving up on phenomenology yet, as without it I never would have undersatood philosophy at all.

Wrt enumeration: In counting to two, you are adding one to one. And even if you are correct that in counting to two you are also doing a subtraction of 1 from 3, this is still mathemtics.

Bearing this is mind, if counting is just doing mathematics, then it seems to me that it should not be surprising that 0 is counted differently. What is less suprising than the fact that 0 is different mathematically than 1!?

 

[118118]

I've googled and skimmed a few bits of text on thin realism. Its "thin" because numbers do not have any properties not ascribed to them by mathematicians. E.g. they are acausal, do not have mass, etc. but I don't see how this is any different to full realism: what properties do full realist ascribe that thin realists do not.

I wasn't 100% satisfied with the idea that they have no explanation of how we acquire mathematical truths other than the working of mathematics. The explanation just terminates there, and I'm not used to explanations stopping like that (I find it unsatisfactory in science, tbh)

Am I right that thin realists only differ from realists in athat they do not ascribe to realism in truth, but only realism in ontology?

I admit to be a bit confused that they often compare maths to astrology. I just don't see science working if the same entities were being referred to and yet 1+1=0. Go and replace all the 1's in scientific study with 2's, what do you think would happen? And if you told the scientists that you had done so, wouldn't the action just amount to using a different symbol, and not actually changing the referrent of 1+1=2 at all.

 

[Jonti]

... I can't raelly say what I believe the subject of maths is though. ...

Would you accept that maths is "the study of form"?

 

[118118]

Well, I'm not raelly sure if many philosophers of mathemtics are real Platonists, if thats what you mean.

 

[Jonti]

No, more that math is all form and no content, iyswim.

 

[118118]

Do you mean content as in matter? So form like, organization.

My oxford companion to philosophy (btw I recommend anyone to get for christmas) says that pure form cannot exist independent of matter, so even if maths was all form it would still depend on matter.

How would we access pure forms?

 

[Jonti]

I really liked this ...

Here's a point of view from Luitzen Brauer where 2 seems to be fundamental:

"Mathematics arises when the subject of two-ness, which results from the passage of time, is abstracted from all special occurrences. The remaining empty form [the relation of n to n+1] of the common content of all these two-nesses becomes the original intuition of mathematics and repeated unlimitedly creates new mathematical subjects." (quoted in Kline 1972, pp. 1199-2000)

He's saying that form is the original intuition of mathematics; and that it is an abstraction from the (perception of the) passage of time.

It may be that, in the absense of matter, the passage of time would not occur (or that it would be meaningless), so I think it's fair to say Brauer's form cannot exist unless matter also exists. But his notion of form is, strictly speaking, not itself dependent on matter, but only upon a phenemenological distinction being possible between two moments of time. If there's a perceptual world of whatever sort, then form, the stuff of mathematics, can be intuited.

 

[Knotted]

I really liked this ...


He's saying that form is the original intuition of mathematics; and that it is an abstraction from the (perception of the) passage of time.

It may be that, in the absense of matter, the passage of time would not occur (or that it would be meaningless), so I think it's fair to say Brauer's form cannot exist unless matter also exists. But his notion of form is, strictly speaking, not itself dependent on matter, but only upon a phenemenological distinction being possible between two moments of time. If there's a perceptual world of whatever sort, then form, the stuff of mathematics, can be intuited.

Its Brouwer not Brauer by the way. I can't even spell when the word is right in front of me.

I must admit to being puzzled about how the passage of time relates to 'two-ness', but I like the idea that 'two-ness' is central to mathematics. I think he regarded time as a synthetic a priori notion and mathematics derives its synthetic a priori existence from this (cf Kant). I see him as being the opposite of the logicists who would see mathematics as being analytic a priori.

Brouwer is, of course, the founder of intuitionism and constructionism in general. See:
http://plato.stanford.edu/entries/mathematics-constructive/
Essentially all mathematics should be constructed from notions in finite mathematics - there's no room for counterfactual reasoning, so the law of the excluded middle is denied.

 

[Jonti]

I think the suggestion is that the (perception of the) passage of time implies two-ness because this is now, and that was then. But, to my mind, it's the perception of difference that's really fundamental. It is only by making a distinction that mathematics become possible; and that implies a cleaving of a previously unified space.

The first distinction, one might say, creates duality.