Are numbers as real as rocks?

Jonti said:

The gods won't help you bob. They invented the integers in the first place

Yeah, amazing, innit? But to write it off as just a coincidence seems a little ... weak.

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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Since integers were invented to represent the way things appear in reality i think your point might need a bit of help there.

A co incidence that a tool designed and refined to simulate reality actually works? oh noes!?

Bob, these are *your* words from #57 ...

Bob_the_lost said:

It's an amazing coincidence that it works so well.

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You appear to be mocking your own point.

Jonti said:

Bob, these are *your* words from #57 ... You appear to be mocking your own point.

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Because the first post was to be read with a certain amount of sarcasm. I'm sorry you missed it.

Numbers were invented to represent reality, thus using the correlation between mathematical modeling and reality to prove that numbers are somehow real is flawed.

Could Dr Johnson have invited Bishop Berkeley to kick a number?

Bob_the_lost said:

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.

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Bob_the_lost said:

Because the first post was to be read with a certain amount of sarcasm. I'm sorry you missed it.

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I can only read what you post. What did you mean by the part I've emphasised?

Jonti said:

I can only read what you post. What did you mean by the part I've emphasised?

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When it's found that mathematics does not explain things then it's altered so that it does. So it's reality (that) changes.

Read up on Newtonian physics for an example of how theory that works is not always true.

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That bit by the looks of things. Or

Since integers were invented to represent the way things appear in reality

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Your arguement is even more flawed than Peewee Herman in a boxing ring with Mike Tyson. You're trying to use the same test scenario as the ones you used to train your model. (If you understand information modeling then this will make sense, otherwise it may require some thought).

Ie mathematics was formulated to simulate reality, because it's been done well does not make it right, true, real or anything other than an invention.

You're just claiming what you want to assert, but not making much of an argument for it.

Bob_the_lost said:

Since integers were invented to represent the way things appear in reality

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That is exactly what Kronecker denied. This thread is full of people more or less agreeing with Kronecker's take and giving their reasons for that.

You're saying "Well, we make up arithmetic to agree with reality, so where's the surprise?" I think the answer to that is, show me an arithmetic with a different set of primes.

(At a pinch, I guess an existence proof will do.)

And you have failed to argue at all. So

Why do you want a different number set with a different set of primes? What do you think that will show?

As for full of people i don't think so. Since the first few posts at least are either outright disagreement and/or derision i think you're on thin ice with that statement. After the first page or so the focus of the thread shifted slightly but even so the majority do not seem to agree with you.

Readers can see for themselves that I've given quite a few reasons for agreeing with Kronecker, as have other people. But folk can read the thread and count these contributions for themselves, if they think it matters.

Bob_the_lost said:

Why do you want a different number set with a different set of primes?

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The thinking here is that inevitable natural principles are "discovered" rather than "invented". So, if there is no other way that the number system could be, it's better described as "discovered".

Bob_the_lost said:

No, that's not -5 at all. FFS did you not read my post?

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.

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Hello Bob.
Seeing absence is a very notmal existential phenomenon
Lol. You are an idealist, which is far more often thought of as delsuional.
I study HPS, BOB, so fuck off telling me to read about Newton
If it is the threory that represents the world, is it just coincidence? But thats not even what you are arguing for - you are arguing for convention right. Would not the invention (!) that 1+1=3 result in a change so fundamental that rocks would no longer be rocks anymore?

I see that most of the force behind your points is that you have calmly mentioned you have some kind of maths degree Well done bob, but philosophers know whats real better than mathematicains, generally speaking at least.

You could recognise that you only have 3 pigs left instead of eight but it's impossible to see -5 pigs.

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No I don't see this.

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.

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Your not seeing the results of Santa Clause, you are seeing -5.

Bob_the_lost said:

Dear gods, they all share the same delusion and therefore they are all right?

Are you on leave from the special school of logical arguement?

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Yeah, erm, you think that 1+1=2 is a delusion?! Why, what do you make it?! I think this post of your doesn't make sense.

Yeah, the argument is actually the real main reason that people are not mathematical idealists - according to the Oxfod Companion to Philosophy or what not. All learning disabled, aren't they Bob. Don't you think you've taken this idea that mathematicians are smart a bit too far there?!

So, its by convention that 1+1=2? Yoy haven't outlined a positive accountof what numbers are yet, unless I've missed it. Which you'll be grumpy enough to point out, yeah?

No, that's not -5 at all. FFS did you not read my post?

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Yeah, and it baffled me? You are under the impression, alot like some of the scientists on here, that cos you know the subject you know philosophy. Thats a delusion, or a "fallacy" (that way you don't use an ad homminen). I mean, have you read a single word of published material on the philosophy of meathematics? So why the fuck are you so sure that all your opponents are delusional?! Why the fuck are you so sure. Its, fucking, creepy mate.

Ad hominem (sp?): personal attack - an invalid argumant.

e mathematics was formulated to simulate reality, because it's been done well does not make it right, true

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So are phjysical laws like speed of light not true? As they were formulated to simulate reality and have been succesful.

Maybe this is an epistemological disagreement!
Epistemology: Theory of knowledege.

118118 said:

Hello Bob.
Seeing absence is a very notmal existential phenomenon

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Well if you don't think i'm real i don't see why i'm going to bother replying to you again.

118118 said:

Lol. You are an idealist, which is far more often thought of as delsuional.
I study HPS, BOB, so fuck off telling me to read about Newton
If it is the threory that represents the world, is it just coincidence? But thats not even what you are arguing for - you are arguing for convention right. Would not the invention (!) that 1+1=3 result in a change so fundamental that rocks would no longer be rocks anymore?

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Damn, it seems my point went stright over your head and then managed to hit a nerve doing so. Wierd, didn't think that could go together.

But i'll reitterate, appologies for not replying to each and every one of your diatribes but i think i counted five of them. Just because people belive in something does not make it real, hence the references to santa claus, religion and delusions.

Now, you've still failed to tell me how there is such a thing as -5. Is it a debt or an abscence, or something else? How is that as real as a rock?

Well OK we have an argument over whether numbers represent reality because they were invented to do so or whether numbers represent reality simply because they are real. For me its not a problem. In my view a key aspect of the usefulness of numbers is that they do not represent reality.

I'll demonstrate. If numbers represented reality then two units would not be equal. One something does not equal one something else. We would have a mathematics where:
1 X= 1' (where X= means 'does not equal')
Naturally
1 X= 1''
and
1' X= 1''
and so on.
We could perhaps talk about:
1+1'
but in the interests of realism we would have to describe physically what adding means. Naturally we could come up with several different notions of addition. Without going into +, +' and +'' and so on, we could at least ask whether our notion of addition is associative and commutative. That is, does:
1+1' = 1'+1
(1+1')+1''=1+(1'+1'') ?
There is no reason why it should of course. If 'addition' is stacking things - putting one on top of the other - then surely it matters what order you stack these things.
So even with just units represented by 1,1' and 1'' we have "several" different examples of "1", "2" and "3". (Incidently we could use our own novel notion of "several" if we prefer).

I can think up several more specific notions of 'number' if this doesn't convince you. In particular I've always been amused by the mathematical distinction between 'less than' or 'less than or equal to' (or finite open sets and finite closed sets if you want to be posh). It has absolutely no phyical interpretation at all.

If a container is full of water below a certain mark then this is exactly the same volume of water if the container is full of water below that mark and including that mark. Physically its a nonsense, but mathematically it is a cornerstone of mathematical analysis.

It also worries me that counting seems to rely on outdated notions of absolute time. Given that counting something is a process that takes a certain amount of time and given that things can decay, fall apart or merge we can't talk about two units seperately from the passage of time. Who says that your notion of now is the real notion of now, and why should different observers agree on the number of objects.

Anyway representing reality is a truly horrible way to start understanding reality. We use proper fiction, which is properly elegant, where we can agree on the conventions. That's number and that's what its about. There seems to be some profound and mysterious reason why we should gravitate to number but saying that numbers are real doesn't do this justice.

The moral is never dismiss jiberish just because its jiberish and never mistake jiberish for god given truth.

Bob_the_lost said:

Well if you don't think i'm real i don't see why i'm going to bother replying to you again.

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With all respect I think that you've misunderstood. I'm not saying that your not real, but that your absence is real too.

Damn, it seems my point went stright over your head and then managed to hit a nerve doing so. Wierd, didn't think that could go together.

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This is the quote that "went over my head"

You can't count - 5 pigs, it's compared to debt as a thought process but it's not physically possible to get hold of -5 pigs the entire idea is stupid.

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Which is the philosophical equivalent to "-5 piggies!!! -5 piggies!!! You can't have -5 piggies". What have I missed?

Just because people belive in something does not make it real, hence the references to santa claus, religion and delusions

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I have never suggested that it does! I have suggested that if something is only true_or_false_by_convention then it is incredible that everyone agrees. Are you suggesting that religion is not objectively false? Or are you agreeing that 1+1 can not = 2 by convention alone?

Now, you've still failed to tell me how there is such a thing as -5. Is it a debt or an abscence, or something else? How is that as real as a rock?

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I would say an absence, but I don't know. I can understand, why you think that rocks are more real, as they are more visceral, you can't get hit on the had with -5pigs. But, come back when you are starving and just lost your last 5 pigs and then tell me that 5 absent pigs are not as real as a rock.

I'm a realist about most things, and as far as I can tell the only other viable alternatives wrt numbers are idealism, or that they are equivalent to fiction - (there are indeed no numbers) - so that there are no prime numbers, there are less than 1000000 prime numbers etc. Thiis seems too sui geneisis, to me, and how would mathematics interact with science if mathematics was only true "in the story"? Would you just put foward a positive account of what numbers are and then I'll try and argue against it.

Sui genesis (sp): totally unique.

Knotted said:

Anyway representing reality is a truly horrible way to start understanding reality.

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Yes. We don't need to represent reality "in our heads" to get the measure of it. It's always at hand to be consulted.

118118 said:

I would say an absence, but I don't know. I can understand, why you think that rocks are more real, as they are more visceral, you can't get hit on the had with -5pigs. But, come back when you are starving and just lost your last 5 pigs and then tell me that 5 absent pigs are not as real as a rock.

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Abscence of pigs is the same as 0 pigs. Not -5 pigs. If you cannot understand the difference between zero and minus five, with all respect due there is no way on earth you should be on a thread discussing mathematics. However neither you or i can see the difference, because you cannot see either one of those states. (except in the most tenuous manner)

Mathematics is a way to describe the world, it has no more reality than that, no intrinsic universal god given existence such as that proposed at the start.

How about an analogy. Language is used to describe the world, it's amazing, pretty much everything has a word or a way of using words to describe it. Therefore language must be somehow "god given". It's just as foolish as an idea as giving integers divine status.

Bob_the_lost said:

Abscence of pigs is the same as 0 pigs. Not -5 pigs. If you cannot understand the difference between zero and minus five, with all respect due there is no way on earth you should be on a thread discussing mathematics. However neither you or i can see the difference, because you cannot see either one of those states. (except in the most tenuous manner)

Mathematics is a way to describe the world, it has no more reality than that, no intrinsic universal god given existence such as that proposed at the start.

How about an analogy. Language is used to describe the world, it's amazing, pretty much everything has a word or a way of using words to describe it. Therefore language must be somehow "god given". It's just as foolish as an idea as giving integers divine status.

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I haven't said that integers are god given And its not amazing that pretty much anything has a word to it - I can easily decide to refer to rocks as jocks, but 1+1 does not legitimately = 3.

Stop repeating your position and give an argument.

Lol at saying I don't know how to minus 5 piggies!

I was having a chat the other day with my psychology lectiurer about how Sartre goes into a Cafe where he is expecting his friend Piere, and the whole e takes on the status of a cafe-without-Piere, or cafe-mius-Pere. You ahve to explain what you mean by "too tenuous", too tenuous for what and why.

What about if your partner had just died, you think that her absence is the same as her never being there. You can feel -5, you can count -5, and Sartre would have that you can see -5. Sure, its a little diferent to +5, but what do you expect, there is a different symbol infront of it.

According to some number scales is not an absence of 5 = to 0 pigs anyway. I really suggest that you do some reading, or outlione a positive account of mathematics - anmd not just repeatedly mumble something about "invention".

PS - imho you are arguing for a fictional account of numbers withouit realizing it. How then, does what is only true "in the stroy" work, in science. If you can answer that question then imho your not ignorant, you just have strong opinions on ontology, depsite what modern philosophy may say otherwise.

And if something is only true_or_false_by_convention why does everyone agree. You poor mathematicians, pressured into agreeing on the most baisc of mathematical problems, when any answer is correct! Mathematics must be objective.

I admit to not reading the post, but

There seems to be some profound and mysterious reason why we should gravitate to number but saying that numbers are real doesn't do this justice.

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Bob_the_lost said:

Language is used to describe the world, it's amazing, pretty much everything has a word or a way of using words to describe it. Therefore language must be somehow "god given"

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I would be more convinced if there was only one legitimate word to describe a given thing. Which there isn't, lots of people disagree what a "rock" is called. Are you seriously suggesting that the only reason you can't write 1+1=3 and get a tick for it is by convention, that you could write a random series of numbers and the only reason you don't get the nobel prize is by convention. Rebel! 1+1 does = 3 by our conventions, and science is not fucked in our hands, oh no!

Jonti said:

Are numbers as real as rocks, but rather more certain?

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Depends, how many rocks?

I don't know if it is relevant but in management speak there is the concept of synergy where 2+2=5 the argument is that if you bring a team together you get more from the teamwork of the team than the sum of the people in the team.

So 2+2=5

weltweit said:

I don't know if it is relevant but in management speak there is the concept of synergy where 2+2=5 the argument is that if you bring a team together you get more from the teamwork of the team than the sum of the people in the team.

So 2+2=5

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I think that's a good point. I've got no problem with addition meaning something like this. Or another example: you have a piece of plastacine and another piece of plastacine and you 'add' them together and hey you have a single (larger) piece of plastacine. So 1+1=1.

Or if we lived in a universe where matter was fragile, homogenous and had strong magnetic forces, we could say that bringing this piece of something together with that piece of something causes this something and that something to divide into many, many small pieces of something. So 1+1=1000 perhaps.

Or we could get pedantic about the groupings we are talking about. Is a speck of dust a rock? I can't see why it isn't, and its pretty difficult talking about rocks without talking about the dust scattered on the rocks. So one rock is not one rock its one rock and that speck of dust and that speck of dust and that speck of dust...

It is a convention that 1+1=2 or 2+2=4 albeit a convention with a curious magnetic charm. Confusion is the correct response here btw.

The words used for particular numbers may be conventional -- but the properties of the numbers are not. Even to ask "How many?" is to take the objectivity of the answer for granted. One is one and all alone and ever more shall be so.

Yes, two drops of water added together can make a single drop. But so what? Two rabbits added together can make a dozen. Again, so what?

In the first case, one has still taken an instantiation of "two"; and applied to that instantiation the concept of merging. One has a new operation which changes the two formerly separate things in their objective and phenomological characteristics. In other words, one has added a further process to that of simple addition (that of merging). One simply cannot do that, without assuming the prior concept of number.

If anything, this kind of sleight of hand is even more evident in the rabbits example. One needs to explain why one ends up with more than two rabbits. Again, this explanation is made just because the prior concept of number obliges one to seek to explain the apparent violation of the laws of number.

And of course the explanation is made in terms of a single ovum being ferilised and dividing into two; and dividing again and again. There is hardly a scientific law that can be expressed without number. If science refers to real things, then so does an instantiation of a scientific principle. In this sense, numbers are indeed prior to individual existences.

While we're on the subject of '-5' its interesting to note that it has two meanings.
1) It means "...subtratact 5" - a bit like the end of a sentence like "...on the mat". This is the naturalistic meaning that we learn early in school, although it is not particularly meaningful on its own - we should really talk about 6-5 or something.

2) It means "negative 5". This is its algebraic meaning. This meaning exists for the purposes of algebraic completeness, a bit like the square root of negative one.

The naturalistic meaning is really quite ugly from an algebraic point of view. Subtraction is awkward - its not associative and its not commutative. Its far more algebraically natural to talk about adding negative 5 than it is to talk about subtracting 5. BODMAS is a horrible thing.

However as Bob has pointed out, negative five has no simplistic physical meaning. It has a meaning in certain areas of physics - for example we can talk about positive and negative electrical charge - but its a nonsense to talk about negative sheep. Negative sheep is a quantity of sheep and this quantity of sheep should have measurable properties (just like negative charge), but a debt does not have physical properties, its just a convention that the debtor and debtee agree on.

Anyway, we could say that there are two different starting points for understanding subtraction. Subtraction or adding negatives. The naturalistic versus the elegant. It reeks havoc in maths education in my opinion.

I found Knotted's post #75 most thought provoking.

It's famous that one can have various different geometries which describe the properties of figures composed out of lines and points. The usual geometry is flat, or Euclidean, Geometry. But one can also have curved space geometries as well. It turns out our ordinary Geometry that describes space so well is just one of many possible geometries. This fact was not appreciated until a couple of thousand years after Euclid.

Now geometry is not the same as number, not by a long way. In Geometry, all true statements can be derived from just a few simple statements or definitions, called axioms. Geometry can be axiomatised. Not so the number system. Although arithmetical statements are "true", arithmetic cannot be axiomatised. This irreducible complexity of arithmetic suggests it is more than just an empty symbol game, more than merely conventional. It suggests arithmetic is somehow just there, waiting to be uncovered.

But, as Knotted points out, the number field that we are all familiar with (that of the so-called natural numbers) is not the only mathematical field. Other fields, almost entirely, but not completely, unlike the natural number field, can be devised. Perhaps it would be possible to have a universe which has a rather different type of number? Perhaps. But could any things exist in it? Things as we know them could not, because things as we know them do follow natural arithmetic.

Would it even be possible for any such universe actually to exist? We know how sensitive our present (numerical) cosmological models are with respect to the value of particular physical constants. On the face of it, this sort of thing would be even more of an issue given a different number field entirely.

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.

Knotted 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

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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.

Negative sheep is a quantity of sheep and this quantity of sheep should have measurable properties (just like negative charge), but a debt does not have physical properties, its just a convention that the debtor and debtee agree on.

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

Jonti said:

The words used for particular numbers may be conventional -- but the properties of the numbers are not. Even to ask "How many?" is to take the objectivity of the answer for granted. One is one and all alone and ever more shall be so.

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But what of the possibility that numbers are conventional but their properties aren't? I'll admit that it sounds weak.

Jonti said:

Yes, two drops of water added together can make a single drop. But so what? Two rabbits added together can make a dozen. Again, so what?

In the first case, one has still taken an instantiation of "two"; and applied to that instantiation the concept of merging. One has a new operation which changes the two formerly separate things in their objective and phenomological characteristics. In other words, one has added a further process to that of simple addition (that of merging). One simply cannot do that, without assuming the prior concept of number.

If anything, this kind of sleight of hand is even more evident in the rabbits example. One needs to explain why one ends up with more than two rabbits. Again, this explanation is made just because the prior concept of number obliges one to seek to explain the apparent violation of the laws of number.

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But if number and numeric operations are objective 'things' of some type then we must be able to describe them without reference to number. I'm just showing that you are not forced to use numbers in the usual way.

Jonti said:

And of course the explanation is made in terms of a single ovum being ferilised and dividing into two; and dividing again and again. There is hardly a scientific law that can be expressed without number. If science refers to real things, then so does an instantiation of a scientific principle. In this sense, numbers are indeed prior to individual existences.

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OK. Say I have a different form of addition where 1+1=111 (or '3' as its usually called). The rule is concatenation with an extra unit. So 2+3=6, 1+7=9 etc. etc. Perhaps the extra '1' is an emergent property of addition.

I can translate this arithmetic easily into ordinary arithmetic.
a+b=a+'b+'1
a+'b=(a+b)-
Where +' is ordinary arithmetic and A- is A with a unit removed.
These two arithmetics are equivalent. I can state any scientific principle using this new arithmetic that you can with the old arithmetic. We are not forced to use one or the other.

However you will find that the old arithmetic uses less breath and less ink, at least when it is applied to our world. I think this is the only reason it is used.