Are numbers as real as rocks?

[118118]

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

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

Maybe its my imagination, but that seemed "well presented"! I disagree with your conclusion however: "presentation" if thats what you want to call it, is vital to help another understand your thought processes, whatever they may be, and thats what a debate is. Didn't Plato (?) commit some kind of supposed fallacy (or argue against) in holding that if you can't express something then you have no reason to believe its true.

Personally, I think you are working on hunches, or feelings (what else could this "thinking" be (charges across a neural network - one is not conscious of these) I don't know much about the mind), which ime are fine but are solely for reaching a verbal conclusion.

 

[118118]

Yeah, I would like to add that I think that it is not thought processes, but the statment 'if x then y, x, therefore y', that are justified by the rules of deductive reasoning. Logical laws cannot be laws of thought, precisely because there may be no laws of thought, at all.

"At least three candidates can be put forward (as bearers of truth) statements, sentences, and propositions".

I was told that natural laws are best described as that which makes law statements true or false.

 

[Knotted]

I disagree with your conclusion however: "presentation" if thats what you want to call it, is vital to help another understand your thought processes, whatever they may be, and thats what a debate is.

Very simple demonstration that this is false:

Proof that 3953 is not prime:
3953=59x67

If this is exact enough for you to agree then explain to me my reasoning.
How do you whether I picked the numbers 59 and 67 at random and got lucky?
How do you know whether I used an algorithm?
If so which algoithm did I use?
How do you know whether I used a heuristic?
If so which heuristic did I use?
How do you know that I used an algorithm which is generally incorrect? I could be using false ideas to arrive at a correct conclusion.
How do you know whether I reverse engineered this?

Personally, I think you are working on hunches, or feelings (what else could this "thinking" be (charges across a neural network - one is not conscious of these) I don't know much about the mind), which ime are fine but are solely for reaching a verbal conclusion.

Which is quite appropriate when a conclusion hasn't been reached!

 

[118118]

Double post

 

[118118]

Very simple demonstration that this is false:

Proof that 3953 is not prime:
3953=59x67

If this is exact enough for you to agree then explain to me my reasoning.
How do you whether I picked the numbers 59 and 67 at random and got lucky?

I couldn't care less. That is proof that 3953 is not prime (I assume that the sum is correct). If the OP was "is 3953 prime" then the question is answered. We are not here to debate your emotions, knotted, but the question that is posed. E.g. are numbers real.

I mean, it helps us to understand your hunch as it says whether the conclusion the hunch leads to is true or false. Whether you are smart enough to find primes is utterly irrelavent.

Which is quite appropriate when a conclusion hasn't been reached!

Yes. But, the point being that no-one wants to watch a thread develop for several months to wait and see if what someone has posted is true or false. You can't argue against a hunch, and it would seem that this is what philosophy is - arguments.

I know that you are wrong through my education.

Just, next time you make a post, write "I have only a hunch that it is the case that..." because it seemed like you were often providing an argument, when in fact you were just discussing the general subject area.

 

[118118]

Does this mean that you go through entire books not paying attention to the arguments, but rather deciding on a hunch what is the case. That seems to me to be a bit of a waste.

Eta: I mean, if I was interested in whether you had chosen the number at random, I could just ask.

Eta2: I mean how am I supposed to convince you otherwise, hope that you suddenly like me, and agree.

All you have shown is the apparently utter pointlessness of discussing things with you

 

[Knotted]

I know that you are wrong through my education.

Oh come on, be honest. Your arguments are based at least as much on your feelings and hunches as my are on mine. Its easier to argue logically if you know what's motivating you. Trust me.

 

[Knotted]

Sometimes I wonder if it is possible to do computerised philosophy.

If we accept that philosophy is (or should be) composed of axioms and rules of inference and that we can formulate these axioms and rules of inference, then we could in theory write a philosophy proving programing.

A lot philosophical questions are slippery but simple matters. We aught to be able to answer most of them with 200-300 words if we could only find the right words. Surely computing power is getting great enough to do searches through the different possibilities.

Non of the above sounds right. Hmm.

 

[118118]

Well. I'm not saying that hunches or a bad thing. Or even that one shouldn't act on them on bb.

But if you want to *engage* with another person then you ought to include an argument.

 

[Knotted]

Well. I'm not saying that hunches or a bad thing. Or even that one shouldn't act on them on bb.

But if you want to *engage* with another person then you ought to include an argument.

Well post 213 was a particularly forcful argument even if I have to say so myself. It forced you to admit that someone's reasoning might be irrellevant to the argument they present. So the forcefulness of the latter is not necessarily conected to the lack of wooliness of the former. You can make of that what you will.

This may seem like a diversion from the main topic, but it isn't in my opinion. I'll return to it in few days time. We need to step back to digest what has been said and you need to get on with your reading.

 

[118118]

Well post 213 was a particularly forcful argument even if I have to say so myself. It forced you to admit that someone's reasoning might be irrellevant to the argument they present. So the forcefulness of the latter is not necessarily conected to the lack of wooliness of the former. You can make of that what you will

Sure you might say A because B, and I say "A, but I think because of C". So in that much your reasoning can be irrelenet. And I agree that someone can agree with something on a hunch. But these are just practical tautologies.

None (?) of my previous points have been refuted. If you want people to agree with your position who don't at the moment then either you must rely on them to do so on good will, or show arguments. And wooly arguments are no type of arguments, and only work by confusing the other poster.

Your not going to change my opinion on the idea that argumentation is more than just a varnish. So I guess we just leave this disussion, and you agree to state when, an argument is wooly or based solely on a hunch and has not been proven.

 

[MullahNasrudin]

Numbers are an abstraction. End of argument surely?

 

[Knotted]

Sure you might say A because B, and I say "A, but I think because of C". So in that much your reasoning can be irrelenet. And I agree that someone can agree with something on a hunch. But these are just practical tautologies.

I'll try it again.

I might reason A because B but I might argue A because C like in post 213. B might involve all sorts of hunches or spurious reasoning and C might be a flawless logical argument. Either way B and C are not necessarily identical. See post 213.

None (?) of my previous points have been refuted.

See post 213.

If you want people to agree with your position who don't at the moment then either you must rely on them to do so on good will, or show arguments. And wooly arguments are no type of arguments, and only work by confusing the other poster.

So it is good when an argument is convincing. This is because an argument tries to convince people of things.

Maybe you should publish this towering feat of philosophical reasoning. It is more convincing than Kant's Critique of Pure Reason or Spinoza's Ethics. I can't find a flaw in it. Bet you nobody else can either.

Your not going to change my opinion on the idea that argumentation is more than just a varnish. So I guess we just leave this disussion, and you agree to state when, an argument is wooly or based solely on a hunch and has not been proven.

I have not presented an argument using wooly reasoning or based solely on a hunch.

 

[Knotted]

Numbers are an abstraction. End of argument surely?

That's the short answer.

The long answer involves amongst other things discussing whether there are possible alternatives that are at least as useful.

 

[118118]

I might reason A because B but I might argue A because C like in post 213. B might involve all sorts of hunches or spurious reasoning and C might be a flawless logical argument. Either way B and C are not necessarily identical. See post 213.

I think I've misunderstood what your saying, again.

If you do provide an argument, then sure, thats not a problem. Neither do I necessarily expect everyone to change their pov every time an an argument is refuted: hunches are fine as long as one politiely say "you may be right about B being fallicious, but I still hold A because I am not willing to give it up yet".

So it is good when an argument is convincing. This is because an argument tries to convince people of things.

Maybe you should publish this towering feat of philosophical reasoning. It is more convincing than Kant's Critique of Pure Reason or Spinoza's Ethics. I can't find a flaw in it. Bet you nobody else can either.

Eh?

I don't understand your reference to post 213. First of all, whether something can be backed up with an argument is important to understand any pov. Secondly, why can't I just ask you if you guessed? Thirdly, as someone who doesn't know you that well, I am far more interestsed in what you have to say on a topic than your thought processes on it - its much more difficult to discuss a hunch than a reasoned pov - a bunch of facts that you have a feeling might be related to your hunch may be/may not be, either way if you continue to only post wooly thinking and hunches *your position is irrefutible except on your own whim* - very annoying for someone who does not disagree (note, of course this does not mean that it is not false, thereby being the problem).

 

[118118]

I think there are terminal problems of misunderstanding between us

the forcefulness of the latter (a post) is not necessarily conected to the lack of wooliness of the former (an argument). You can make of that what you will

Personally, I think you are working on hunches, or feelings (what else could this "thinking" be (charges across a neural network - one is not conscious of these) I don't know much about the mind), which ime are fine but are solely for reaching a verbal conclusion.

Which is quite appropriate when a conclusion hasn't been reached!

"presentation" if thats what you want to call it, is vital to help another understand your thought processes, whatever they may be, and thats what a debate is.

Very simple demonstration that this is false

I hate it when 118118 tells me to explain my chain of reasoning, I'm much happier when reasoning isn't in chains

but

I have not presented an argument using wooly reasoning or based solely on a hunch.

 

[118118]

Numbers are an abstraction. End of argument surely?

You have to at least explain why its basic assertions enjoy such a very high degree fo certainty - how can any rational being doubt that 2+2=4 (admittedly a few people on here seem to, but hey). Why does mathemtics seems necessary and a priori, I mean.

 

[Knotted]

I don't understand your reference to post 213. First of all, whether something can be backed up with an argument is important to understand any pov.

A clear argument backs up supposed facts that are being presented. These supposed facts may or may not be the same thing as the arguer's (is there such a word?) point of view. To get someone to understand a point of view can rely on all sorts of communication strategies not just formal polemics.

I think I'm being too exact for you rather than not exact enough.

Secondly, why can't I just ask you if you guessed?

You can, but as you pointed out you do not need to in order to convince yourself of the truth that 3953 is not prime. However the method I use for establishing this truth might be much more important than the truth itself, because it might be a stunningly brilliant method for establishing whether other numbers are primes.

So the method I use and the argument I present might be two different things which might be important in two different ways.

Thirdly, as someone who doesn't know you that well, I am far more interestsed in what you have to say on a topic than your thought processes on it

You might be able to learn something from either. You should be able to criticies either. I don't understand your prejudice against thought.

- its much more difficult to discuss a hunch than a reasoned pov - a bunch of facts that you have a feeling might be related to your hunch may be/may not be, either way if you continue to only post wooly thinking and hunches *your position is irrefutible except on your own whim* - very annoying for someone who does not disagree (note, of course this does not mean that it is not false, thereby being the problem).

I don't only post wooly thinking and hunches. Although I have no problem with either. I also don't understand why it is problematic to someone who does not disagree (agrees surely?) with it. You can ignore it or you can use it, either way its harmless.

Suppose you do a PhD and you collaborate with someone to write an original paper. Think to yourself what sort of skills other than rigorous thinking you would need to do this successfully. Skills relating to formal polemics on their own will not work because apart from the fact that it is an informal collaboration neither you nor the person you are working with has established what you are trying to prove.

 

[Knotted]

I think there are terminal problems of misunderstanding between us

I must admit that this anoys me. I think I can understand you perfectly well and I feel that I'm failing as an explainer if you can't understand me.

This topic is like building a tower in a desert. You have to be able to deal with reasoning on shifting sands. There are no simple basic truths from which to start because we are questioning what are usually regarded as simple, basic truths.

 

[118118]

Ok, we agree. Hunches are fine. The only problem I must have is knowing when something is a hunch that I can simply say "no" to, or there is an argument burried in the post. Enough of this, someone post something on numbers.

 

[Knotted]

Ok here's a strand of thought. Sorry for the delay.

There's two types of work to be done on this question (and I think philosophical questions in general). There is work to be done on the meaning of the terminology used in the question and there is work to be done on the question itself.

The problem with the first type of work is that it is never complete (this is very easily proved by the way). The problem with the second type of work is that it cannot be properly completed until the first type of work is complete.

Who knows what 'realism' really means? Is there a proper metaphysics for it? I think it is obvious that there isn't. Which isn't to adopt an anti-realist position. Its to say that the question has only limited interest.

For example most will agree that physical matter is real. However, some might consider it an abuse of language to describe an object's mass as real. Now that's not to say that the object has does not have a mass its just that the mass of the object cannot be considered seperately from the object itself. Is 1kg real? Is it as real as 1kg of lead? Is it actually more real than 1kg of lead in that it is not just a property of 1kg of lead but also 1kg of feathers? More to the point, are these questions of any interest?

A word like real is does not have a metaphysical meaning. It is an analogy.

Consider a hill. Its a real hill, with wiggly contours and all. A scientist rolls a ball down the hill and after repeated experimentation discovers that the ball tends to follow a line. Is that line real? I would say that it is as real as mathematics in general. Or rather I would say that the same (or at least a very close) notion of 'real' can be applied to that line as can be applied to mathematics.

The scientist is the one who imposes the line on the hill because lines do not physically exist. However the scientist does not pick a line at random and the line could be discovered by any scientist. So it is objective.

Now consider a pure mathematical notion. A definition of prime numbers:

A prime number is a natural number (whole positive number) excluding 1 such that it is divisible only by itself and 1.

The most obvious question is why exclude 1?

There is no profound metaphysical reason why 1 should be excluded. Every theorem of mathematics could be easily restated if the definition of prime was changed. Everytime the "set of primes" is referred to it could be replaced by the "set of primes excluding 1" and everytime the "set of primes or 1" is referred to it could be replaced by the "set of primes". In a similar way there is no metaphysical reason why a ball on the hill is on the line the scientist has observed.

The simple reason that 1 is not considered a prime is that the number of "excluding 1" clauses that would have to be introduced is considerably greater than the "or 1" clauses that could be deleted. Mathematics is much more elegant (simply stated) if 1 is not considered a prime.

OK so does that mean that the notion of prime number is not real? Not really. The last sentence of the above paragraph is I think true in an objective way even if it needs refining and proving - essentially it is information theoretic. So the fact that mathematics is chosen to be such a way that it is sufficiently rich and elegant seems to indicate (I haven't proved this) that it is meaningful to talk about a certain 'realism' to mathematics - in a similar way the path that the ball takes is 'real' even if it is artificially imposed on the hill by the scientist.

Does that resolve the problem?

 

[118118]

The trajectory that a ball takes down a hill, I would assume to be a real event. But the line that one mentally draws is not a real trajectory. So the question is, are numbers real (balnk). What is the blank?

From the little I have read on the subject (a couple of pages, a couple of times), "realism-in-ontology is the view that the subject matter of mathematics (numbers and the like) is a realm of objects that exist independent of the mind concentions and languages of mathematicians"
"realism-in-truth-value is the view that assertions of mathematics are non-vacuously true or false, indepndent of the mind, language and conventions of the mathematician" This ussually implies realism-in-ontology.

I don't know why the anti-realist do not think realism is true. Anyone? The sheer certainty of basic mathematical assertions is the strongest argument for realism, 2+2=4 is undoubtable as there being a rock in front of one.

 

[goldenecitrone]

I asked this question to a load of art historians yesterday evening, but unfortunately I don't think most of them understood the question. Oh, well.

 

[118118]

it is meaningful to talk about a certain 'realism' to mathematics - in a similar way the path that the ball takes is 'real' even if it is artificially imposed on the hill by the scientist

This made me think of Merleau-Ponty and the type of realism that he condoned. As yet I haven't 100% systematized what he's saying, so he still seems to be arguing, in some ways, for a form of dualism, but his views are interesting none-the-less. He argues that every object that we see, e.g. a rock, is interpreted and as such meaningful; and a physical object is never meaningful in itself. As a rock appears to me is not what a rock is in itself. The object of perception in itself has some structure and some properties that can be predicated as a rock, although it exists unpredicated before being preceived; we do not create properties of being a rock at our will, rather the object exists in the external world as a potential object of predication.

So the question I am asking myself, Knotted, is how is this any different from rocks?

Incidently, Merleau-Ponty thought that the source of mathematical knwoledge is empirical. This is not a popular position, at all, though Wittgenstein did take the trouble to criticise it.

 

[118118]

I guess one could ask are numbers as real as matter. Matter being the paradigm of real, that everything is made out of, would make that quite difficult.

 

[Jonti]

I asked this question to a load of art historians yesterday evening, but unfortunately I don't think most of them understood the question. Oh, well.

Well, when I asked one, I got the reply ...

"We take as given the idea of distinction and the idea of indication, and that one cannot make an indication without drawing a distinction. We take therefore the form of distinction for the form."

I think that implies a sort of realism

 

[Knotted]

From the little I have read on the subject (a couple of pages, a couple of times), "realism-in-ontology is the view that the subject matter of mathematics (numbers and the like) is a realm of objects that exist independent of the mind concentions and languages of mathematicians"
"realism-in-truth-value is the view that assertions of mathematics are non-vacuously true or false, indepndent of the mind, language and conventions of the mathematician" This ussually implies realism-in-ontology.

If we assume that both these types of realism are correct and we assume that objects can be enumerated then we run into terminal difficulties very quickly.

We can enumerate numbers like 4. We can talk about 3 4's and 5 4's and we add them together to get 8 4's as 3+5=8. In full 3x4+5x4=(3+5)x4=8x4.

But can we enumerate 0's? Effectively no. 3 0's and 5 0's are still 8 0's [ie. the distributive law still operates] but 3 0's and 5 0's are also 9 0's or 2 0's or 0 0's etc. If there are '0' objects then we cannot distinguish between them so when talking about 0 as an object we have the mathematical law 1+1=1.

Set theoretic notions of collections and collections of collections and collections of collections of collections etc. also has run into problems when the sets are defined in a purely conceptual manner. There are various set theoretic paradoxes, the most famous being Russell's paradox.
http://en.wikipedia.org/wiki/Russell_paradox

How do mathematicians talk about large collections of sets while avoiding these types of paradox? Simply by talking about classes of sets rather than sets of sets. So in mathematics there are at least two types of grouping together of things. But how do we collect classes together and if we can't is it meaningful to count them? If we consider the class of all sets containing the empty set and the class of all sets not containing the empty set are these 'two' classes really two objects?

 

[Knotted]

This made me think of Merleau-Ponty and the type of realism that he condoned. As yet I haven't 100% systematized what he's saying, so he still seems to be arguing, in some ways, for a form of dualism, but his views are interesting none-the-less. He argues that every object that we see, e.g. a rock, is interpreted and as such meaningful; and a physical object is never meaningful in itself. As a rock appears to me is not what a rock is in itself. The object of perception in itself has some structure and some properties that can be predicated as a rock, although it exists unpredicated before being preceived; we do not create properties of being a rock at our will, rather the object exists in the external world as a potential object of predication.

Yes I think I agree with this. Obviously things aren't meaningful in themselves, meaning is at least partially associated with the subject. And I certainly agree that our perceptions of things are not the same as the things themselves. I'm not sure about the predicates part though.

So the question I am asking myself, Knotted, is how is this any different from rocks?

To clarify are you asking, "how is the reality of an abstract line associated with a real experiment different to the reality of a rock?"

The obvious answer is that rocks have a material existence whereas abstract lines do not. You mioght use the word 'real' in blunt, clumsy sort of way to describe both if you want but you will have to recognise different shades of 'realism' if you do.

An answer with respect to Merleau Ponty reference I'm guessing would go along the lines of:
The line in question is not an object but a predicate. Whereas the rock is an object but not a predicate.

I should add the disclaimer that the above is the guess at the answer for somebody who I have never read to a question that I'm not clear on!

Incidently, Merleau-Ponty thought that the source of mathematical knwoledge is empirical. This is not a popular position, at all, though Wittgenstein did take the trouble to criticise it.

I would disagree with Merleau-Ponty here.

 

[118118]

Whereas the rock is an object but not a predicate.

No, he would say that the rock is a predicate (like blue), its brute matter that is not.

 

[118118]

If we assume that both these types of realism are correct and we assume that objects can be enumerated then we run into terminal difficulties very quickly.

We can enumerate numbers like 4. We can talk about 3 4's and 5 4's and we add them together to get 8 4's as 3+5=8. In full 3x4+5x4=(3+5)x4=8x4.

But can we enumerate 0's? Effectively no. 3 0's and 5 0's are still 8 0's [ie. the distributive law still operates] but 3 0's and 5 0's are also 9 0's or 2 0's or 0 0's etc. If there are '0' objects then we cannot distinguish between them so when talking about 0 as an object we have the mathematical law 1+1=1.

Set theoretic notions of collections and collections of collections and collections of collections of collections etc. also has run into problems when the sets are defined in a purely conceptual manner. There are various set theoretic paradoxes, the most famous being Russell's paradox.
http://en.wikipedia.org/wiki/Russell_paradox

How do mathematicians talk about large collections of sets while avoiding these types of paradox? Simply by talking about classes of sets rather than sets of sets. So in mathematics there are at least two types of grouping together of things. But how do we collect classes together and if we can't is it meaningful to count them? If we consider the class of all sets containing the empty set and the class of all sets not containing the empty set are these 'two' classes really two objects?

I'm not sure what remains of your point when it is insisted that enumeration is mathematics. Therefore, as 0+0=1 is not mathematics, 0+0=1 is not enumeration either. 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.

1+1=1 is just untrue.