0
   

Best argument for Platonism.

 
 
TuringEquivalent
 
  1  
Reply Fri 9 Jul, 2010 06:19 pm
@ughaibu,
ughaibu wrote:

Which he did within a strictly formalised system. One problem is that Platonism is, as far as I can see, incompatible with Godel's method (see below).


I hope you give me more detail than this!

Quote:
The problem for realists is the implication that some abstract objects both exist and dont exist.


Why/How?


Quote:

TuringEquivalent wrote:
i believe in laws of nature.
I know, but your beliefs dont define my reality.



You misunderstood. It is a technical matter. I believe in laws of nature in short means:

1. Laws of nature are dispositional properties of natural kinds.
or
2. Nomic necessary in nature.


My point is that you can eliminate God( something very metaphysical) , but you can` t eliminate the metaphysics. Both 1, and 2 are metaphysical theories .

Quote:
Basically, Platonists are taking the piss. If they claim that false propositions exist as abstract objects, then the claim of correspondence fails. The object 1+1=3 exists, therefore, by correspondence, the proposition is true. So the problem is referred, they need second order abstract objects, but then face the same problem. So, this is not a solution to any problem and they need to support the existence of uncountable infinities of objects. I dont see how they can do the latter, do you? If you cant, then the Platonist position commits you to constructivism.

+ rest


Ok, the basic theory is that names have reference. So, the name '3' corresponds to a AO, 3. Platonists commit to the property that 3 is prime, so, the proposition '3 is prime' is true, and '3 is not prime' is false. The latter do not correspond to any AO. So, it is not right to say " false proposition exist as AO". The AO of '3 is prime' is 3, and not '3 is prime', or ' 3 is not prime'.

TuringEquivalent
 
  1  
Reply Fri 9 Jul, 2010 07:01 pm
@mickalos,
mickalos wrote:


First of all, I don't know why you would want to separate (1) and (3). Where would platonic entities exist if not in the mind or the world? Assuming that by the world, we do not mean simply Earth, and in fact we mean everything that is the case, then platonic entities exist in the world. Where else are they to exist if not in reality? Perhaps by (1) you mean nominalism.


By 1, i mean the our actual concrete world. Normally, philosophers distinguish our concrete world, where each parts have spatial-temporal relation with one another, to abstract objects.


Quote:

However, far more importantly, my position is not that mathematical objects exist in the mind. I have no idea how you could have possibly inferred that from what I have said.


I never claim that.


Quote:
My view is that there is no such thing as mathematical objects: numbers, propositions, operations, you name it. Platonism is founded upon a representational model of language which I reject; language behaves neither as a mirror of the world, nor as a mirror of mental entities in the mind (what a mental entity is even supposed to be, I could not say). When I perform calculations, just as when I greet somebody with a "Hello", or give somebody the middle finger, I do not correctly or incorrectly represent the world, the mind, the platonic heavens, or anything else. I am engaging in a practice, I exhibit technical skill and efficiency in my practices, but I represent nothing, and the question of my words corresponding to things simply does not arise..


I already know this was your view before. Perhaps what you have in mind is something like a language game from Wittgenstein. If mathematics is grounded in practice, or part of a language game, it can` t reflect the objectivity of mathematics as being " true in all possible worlds".

Quote:
Quote:


I said you were bias toward AO, and favor concrete objects. You have to deal with your own personal demons.

No, the concept of an abstract object in the metaphysical sense in which platonists talk, i.e. of things that actually exist, is meaningless. I do not mean 'meaningless' in a merely pejorative sense, expressing my own prejudices, I mean that the term is quite literally meaningless in the same sense as, "Caesar is a prime number", "I see an immaterial chair", "The nothing nothings" (Heideggar), "The gulf war took place in a hyperreality" (Baudrillard), etc.



I know what you mean, and you are simply wrong. Abstract objects and their properties are posited by platonist. The AO 3 has the property of being prime, and so '3 is prime' is true. If you doubt AO exist, then you simply don` t know AO is a posit. If you doubt AO 3 has the property of prime, then you simply don ` t know the property of AO 3.
Quote:


Certainly, in order to be a platonist, or at least behave like one, all you need to do is say that '3 is prime' corresponds to some abstract object, but in order to justify yourself (not only to others, but one should also be able to satisfy oneself in ones beliefs), or show your belief to be true, you need to give reasons, evidence and proof, and you need to defend yourself against challenges posed. To me, it sounds as if you are simply accepting platonism as a matter of faith.


The justification for platonism is quite obvious. Platonism explains all the facts we want in a theory that explains the objectivity of mathematics. It is also the best view compare to all the rivaling implausible views.


Quote:
Surely you don't want to say that there is more than one number that satisfies 2+2=x? Certain polynomial equations have more than one answer, but not 2+2.


Well, there is a famous paper in the field that said one can look at the integers as sets. The author show there are many ways to construct these sets. This is why the author of "platonism, and anti platonism" said there need not be any uniqueness. This is consistent with '3 is prime' corresponds to only one AO 3.


Quote:
On closer reading, I think perhaps Emil was talking about the indeterminacy of reference. One argument in favour of this: On one model, M, we might map "3 is prime" to a an abstract object, AO1, and map "3 is odd" to AO2. Imagine we now reverse the mapping so that we map "3 is prime" to AO, and map "3 is odd" to AO1, and call this M*. Now, in M and M* both "3 is prime" and "3 is odd" is true, but they represent the platonic heavens differently, which one is right? All of our observations and behaviour are compatible with either.


This is not a problem as i said before.
ughaibu
 
  1  
Reply Sat 10 Jul, 2010 03:00 am
@TuringEquivalent,
TuringEquivalent wrote:
ughaibu wrote:
The problem for realists is the implication that some abstract objects both exist and dont exist.
Why/How?
Because incompatible theorems can be proved, within mathematics as a whole.
TuringEquivalent wrote:
The latter do not correspond to any AO.
The problem is that if you need an abstract object in order to make something true, then you need some way to make things false. Simply not being true is insufficient, otherwise nonsensical statements can be used in mathematics. And being bereft of false statements rules out proofs by contradiction, leaving the Platonist claim restricted to constructive maths. Accordingly, Platonists will need to adopt the position that classical maths, which employs false statements, is fiction. Platonism reduces to fictionalism for classical maths.
TuringEquivalent
 
  1  
Reply Sat 10 Jul, 2010 03:38 am
@ughaibu,
ughaibu wrote:

TuringEquivalent wrote:
ughaibu wrote:
The problem for realists is the implication that some abstract objects both exist and dont exist.
Why/How?
Because incompatible theorems can be proved, within mathematics as a whole



This is not clear to me. If you are talking about the same deductive system generating two contradictory theories, then this is not a consistent system, and thus, a useless tool. Perhaps, you mean two different deductive system with different axioms. If so, this is consistent with platonism, since, these two deductive system describes different classes of AOs.

Quote:

The problem is that if you need an abstract object in order to make something true, then you need some way to make things false.


We know ' 3 is blue' is falses, because the predicate 'is blue' simply do not apply to the AO 3. In this example, i need to know the AO 3, and i need to know the predicate ' is blue' does not to AO 3. what can we conclude? To figure out if Fn is false, i just need to know the predicate F does not apply to the object, AO n .


ughaibu
 
  1  
Reply Sat 10 Jul, 2010 03:50 am
@TuringEquivalent,
TuringEquivalent wrote:
this is consistent with platonism, since, these two deductive system describes different classes of AOs.
Presumably Platonists think that something like the axiom of choice corresponds to exactly one abstract object, yet the axiom of choice can be true or false.
TuringEquivalent wrote:
To figure out if Fn is false, i just need to know the predicate F does not apply to the object, AO n .
But, in the case of a proof by contradiction, you will just have meaningless statements.
kennethamy
 
  1  
Reply Sat 10 Jul, 2010 04:32 am
@TuringEquivalent,
TuringEquivalent wrote:

ughaibu wrote:

TuringEquivalent wrote:
ughaibu wrote:
The problem for realists is the implication that some abstract objects both exist and dont exist.
Why/How?
Because incompatible theorems can be proved, within mathematics as a whole



This is not clear to me. If you are talking about the same deductive system generating two contradictory theories, then this is not a consistent system, and thus, a useless tool. Perhaps, you mean two different deductive system with different axioms. If so, this is consistent with platonism, since, these two deductive system describes different classes of AOs.

Quote:

The problem is that if you need an abstract object in order to make something true, then you need some way to make things false.


We know ' 3 is blue' is falses, because the predicate 'is blue' simply do not apply to the AO 3. In this example, i need to know the AO 3, and i need to know the predicate ' is blue' does not to AO 3. what can we conclude? To figure out if Fn is false, i just need to know the predicate F does not apply to the object, AO n .





You seem to be confusing the predicate, "is not true" with the predicate, "is false". It does not follow from the premise that it is not true of the number 3 that the number 3 is blue, that it is false that the number 3 is false.
TuringEquivalent
 
  1  
Reply Sat 10 Jul, 2010 05:36 am
@ughaibu,
ughaibu wrote:

TuringEquivalent wrote:
this is consistent with platonism, since, these two deductive system describes different classes of AOs.
Presumably Platonists think that something like the axiom of choice corresponds to exactly one abstract object, yet the axiom of choice can be true or false.


It is consistent for platonists to hold that axioms are descriptions of a class of OA. As for any descriptions, it can either be true, or false depending on the description.

Quote:

But, in the case of a proof by contradiction, you will just have meaningless statements.


This is really technical details for mathematicians, but are completely not relevant for philosophers. For philosophers, the concern is semantics about how to interpret Fn. Platonism, formalism etc are all attempt to answer how we might interpret Fn.
TuringEquivalent
 
  1  
Reply Sat 10 Jul, 2010 05:42 am
@kennethamy,
kennethamy wrote:

TuringEquivalent wrote:

ughaibu wrote:

TuringEquivalent wrote:
ughaibu wrote:
The problem for realists is the implication that some abstract objects both exist and dont exist.
Why/How?
Because incompatible theorems can be proved, within mathematics as a whole



This is not clear to me. If you are talking about the same deductive system generating two contradictory theories, then this is not a consistent system, and thus, a useless tool. Perhaps, you mean two different deductive system with different axioms. If so, this is consistent with platonism, since, these two deductive system describes different classes of AOs.

Quote:

The problem is that if you need an abstract object in order to make something true, then you need some way to make things false.


We know ' 3 is blue' is falses, because the predicate 'is blue' simply do not apply to the AO 3. In this example, i need to know the AO 3, and i need to know the predicate ' is blue' does not to AO 3. what can we conclude? To figure out if Fn is false, i just need to know the predicate F does not apply to the object, AO n .





You seem to be confusing the predicate, "is not true" with the predicate, "is false". It does not follow from the premise that it is not true of the number 3 that the number 3 is blue, that it is false that the number 3 is false.


Are 'is not true', and 'is false' predicates at all? Normally, predicates tell us something about properties, while truth values tell us something completely different. So 'cat is blue' is true tell us something about the cat, and the predicate 'is blue' tell using something about the properties.
0 Replies
 
ughaibu
 
  1  
Reply Sat 10 Jul, 2010 05:44 am
@TuringEquivalent,
TuringEquivalent wrote:
It is consistent for platonists to hold that axioms are descriptions of a class of OA. As for any descriptions, it can either be true, or false depending on the description.
This is really technical details for mathematicians, but are completely not relevant for philosophers. For philosophers, the concern is semantics about how to interpret Fn. Platonism, formalism etc are all attempt to answer how we might interpret Fn.
Okay, you haven't convinced me that Platonism is more plausible and less counter-intuitive than the alternatives, and as your argument seems to be something like "wouldn't it be nice if. . . . so let's pretend. . . ", I continue to reject Platonism.
One point, left over from the other thread, if abstract objects are required for mathematical statements to be true, then they're required for any generalisation to be true. Take the statement "there exists a causally effective agent", if this statement requires an abstract object there is an abstract object which is a causally effective agent. But all abstract objects are causally inert, so this is impossible. I think most people would choose to reject abstract objects rather than be committed to the claim that there are no causally effective agents. Accordingly, I think Platonism is highly implausible.
mickalos
 
  1  
Reply Sat 10 Jul, 2010 07:14 pm
@TuringEquivalent,
TuringEquivalent wrote:

I already know this was your view before. Perhaps what you have in mind is something like a language game from Wittgenstein. If mathematics is grounded in practice, or part of a language game, it can` t reflect the objectivity of mathematics as being " true in all possible worlds".


A language game is exactly what I have in mind. Of course, for Wittgenstein, a language game is not merely the exchange of words; language is interwoven with our activities and behaviour. In short, language is part and parcel of our practices, which is partly why Wittgenstein introduces the term language-game rather than simply using the term 'language'.

You are talking about necessity rather than objectivity. It should be clear that necessity is an even bigger problem for the platonist. Platonists construe the subject matter of mathematics as objects of some description; how on earth are we to understand the notion that any given platonic entity that corresponds to a number necessarily exists, and necessarily has certain properties? This clearly brings out why modelling mathematical facts on descriptive facts creates a confused chimera of a theory. The notion of a table having necessary existence does not make sense, nor does it make sense that a ball could be necessarily red. Yet the Platonist claims that this is exactly the case with abstract objects, that somehow this logical necessity exists over and above our practices, but offers only mysticism as an explanation. In short, platonism does not offer an account of logical necessity.

As we have already discussed, the Wittgensteinian viewpoint is that nothing compels us to put 4 after 2+2=..., just as nothing forces Lewis Carroll's tortoise to infer Q from "P, and if P, then Q". Calculating is a practice, it involves the application and mastery of a technique, and nothing to do with discovering mythological objects in the heavens. But, surely our practices might have been different; how then, is the hardness of the logical must to be accounted for? The Wittgensteinian response, in a sense, is to reconstruct our misleading notions of necessity along naturalistic, Quinean lines. It is best to lift it straight from Wittgenstein I think:

""But mathematical truth is independent of whether human beings know it or not!"—Certainly, the propositions "Human beings believe that twice two is four" and "Twice two is four" do not mean the same. The latter is a mathematical proposition; the other, if it makes sense at all, may perhaps mean: human beings have arrived at the mathematical proposition. The two propositions have entirely different uses.—But what would this mean: "Even though everybody believed that twice two was five it would still be four"?—For what would it be like for everybody to believe that?—Well, I could imagine, for instance, that people had a different calculus, or a technique which we should not call "calculating". But would it be wrong? (Is a coronation wrong? To beings different from ourselves it might look extremely odd.)"

In other words, certainly, our practices might have been different; we might have constructed for ourselves a completely different form of life into which the notions of numbers and mathematics have no place or application. This, however, properly understood, is simply to say that we would have had different practices, practices different from our actual practices of calculating. By way of an analagy, might the rules of chess have been different? Well, certainly, we might have played a different game.

I, personally, take a slightly more liberal view than the reading of Wittgenstein suggested above (although, it is more than plausible that he agrees with me). I would say, that in certain circumstances, it would be right to call different practices from our actual ones "calculating". Clearly, were people to have a different calculus such that 2+2=5 and a whole raft of other results, they would have to have different practices that related to these; for, if they simply said that 2+2=5 and went on in exactly the same way we do, we should simply say they were speaking a different language, but had the same mathematics. However, it is conceivable that these different practices might be very similar to ours, for example, if their calculus was still used to build bridges, measure distances, and derive theoretical results. Clearly, the mathematical techniques would have to be applied, and would result in different consequences, but with sufficient similarity to our actual practices, there seems to be no problem with saying, "They calculate differently to us." Imagine, for example, we made contact with a tribe that played a board game identical to our own chess, except without the bishops. Do they have a radically different practice, or do they merely have a "different form of chess"?

Another important point against the platonist account of mathematical necessity, hinted at in Wittgenstein's words above, is that it leaves open the possibility that we have always been wrong in our belief, for example, that 2+2=4. The platonist says that mathematical entities stand in an eternal relation to one another in the platonic heavens, independently of whatever we believe of them, thus, we may one day discover that all of human kind has always been wrong about 2+2=4, and that in fact, this relation does not hold. This seems utterly absurd to me.

Quote:

I know what you mean, and you are simply wrong. Abstract objects and their properties are posited by platonist. The AO 3 has the property of being prime, and so '3 is prime' is true. If you doubt AO exist, then you simply don` t know AO is a posit. If you doubt AO 3 has the property of prime, then you simply don ` t know the property of AO 3.

You are espousing a form of constructivism, here. Platonists do not posit anything, platonists say that mathematical objects exist, independently of human mental states. Platonism has become practically synonymous with realism, clearly, if something is merely "posited", it is not real.

Quote:

The justification for platonism is quite obvious. Platonism explains all the facts we want in a theory that explains the objectivity of mathematics. It is also the best view compare to all the rivaling implausible views.

As I have quite clearly demonstrated throughout this thread, platonism offers no explanations. Its explanation of necessity is that objects stand in eternal relations to one another; its explanation of the subject matter of mathematics is abstract objects that actually exist. These are not answers, they are mystical babblings. What is an eternal relation? What is an abstract object? Where do they they exist? How do I know about them? How do my words refer to them? Platonism explains nothing. Moreover, the idea of shadowy objects that somehow necessarily exist, in a "place" without spacio-temporal properties, that we apparently know about but we do not know how we know about, is not merely implausible, it is nonsensical.


Quote:
Well, there is a famous paper in the field that said one can look at the integers as sets. The author show there are many ways to construct these sets. This is why the author of "platonism, and anti platonism" said there need not be any uniqueness. This is consistent with '3 is prime' corresponds to only one AO 3.
Quote:
On closer reading, I think perhaps Emil was talking about the indeterminacy of reference. One argument in favour of this: On one model, M, we might map "3 is prime" to a an abstract object, AO1, and map "3 is odd" to AO2. Imagine we now reverse the mapping so that we map "3 is prime" to AO, and map "3 is odd" to AO1, and call this M*. Now, in M and M* both "3 is prime" and "3 is odd" is true, but they represent the platonic heavens differently, which one is right? All of our observations and behaviour are compatible with either.


This is not a problem as i said before.

So now you are saying that numbers are sets? Presumably, the only way that this view can be consistent with platonism is if sets are themselves platonic, or you are saying that the sets express integers, but refer to platonic entities. Now, depending on how you define 0 and the the successor function, 2 can be constructed as, {{ }, {{ }}}, or with other definitions, it could be constructed as {{{ }}}. My knowledge of set theory is pretty rudimentary, but as I understand it these are two different sets (correct?). If you are committed to the view that numbers are sets and that sets are platonic entites, then you are committed to the view that there is not a number two, but there are in fact two number twos. Sometimes we as the first number two to the second number two to get one of the many number fours? In essence, you would be committed to saying that '2+2=4' can be used to express a number (presumably an infinity) of different true propositions. Clearly, this is absurd, and in any case, you still have the problem of how the signs {{ }, {{ }}} and {{{ }}} refer/map on to the platonic sets.

Likewise, if you are saying that these different constructions all refer to the same platonic entity, 2, the problem of indeterminacy of reference remains. All of our observations, behaviour and results are compatible with a number of different ways of mapping our signs onto mathematical reality. Platonism tells us that there is only one correct mapping, yet there is no fact of the matter that determines which mapping is correct.


TuringEquivalent
 
  1  
Reply Sat 10 Jul, 2010 08:19 pm
@ughaibu,
ughaibu wrote:

TuringEquivalent wrote:
It is consistent for platonists to hold that axioms are descriptions of a class of OA. As for any descriptions, it can either be true, or false depending on the description.
This is really technical details for mathematicians, but are completely not relevant for philosophers. For philosophers, the concern is semantics about how to interpret Fn. Platonism, formalism etc are all attempt to answer how we might interpret Fn.
Okay, you haven't convinced me that Platonism is more plausible and less counter-intuitive than the alternatives, and as your argument seems to be something like "wouldn't it be nice if. . . . so let's pretend. . . ", I continue to reject Platonism.
One point, left over from the other thread,


What did i do? I think what i did was to correct your misunderstanding with platonism. I don`t think you understand the basics well enough to understand why platonism is better. You make very very simply mistakes like:
Quote:

if abstract objects are required for mathematical statements to be true,then they're required for any generalisation to be true.


This is completely ridiculous. Yes, i have the benefit of a philosophical, and mathematical training, but do you honestly don`t see problem with this statement? The "they" in this statement refers to AO, but statements like 'cat is in the mat' is determined by the concrete object, cat, and not an AO.

TuringEquivalent
 
  1  
Reply Sat 10 Jul, 2010 09:06 pm
@mickalos,
mickalos wrote:

TuringEquivalent wrote:

I already know this was your view before. Perhaps what you have in mind is something like a language game from Wittgenstein. If mathematics is grounded in practice, or part of a language game, it can` t reflect the objectivity of mathematics as being " true in all possible worlds".


A language game is exactly what I have in mind. Of course, for Wittgenstein, a language game is not merely the exchange of words; language is interwoven with our activities and behaviour. In short, language is part and parcel of our practices, which is partly why Wittgenstein introduces the term language-game rather than simply using the term 'language'.

You are talking about necessity rather than objectivity. It should be clear that necessity is an even bigger problem for the platonist. Platonists construe the subject matter of mathematics as objects of some description; how on earth are we to understand the notion that any given platonic entity that corresponds to a number necessarily exists, and necessarily has certain properties? This clearly brings out why modelling mathematical facts on descriptive facts creates a confused chimera of a theory. The notion of a table having necessary existence does not make sense, nor does it make sense that a ball could be necessarily red. Yet the Platonist claims that this is exactly the case with abstract objects, that somehow this logical necessity exists over and above our practices, but offers only mysticism as an explanation. In short, platonism does not offer an account of logical necessity.

As we have already discussed, the Wittgensteinian viewpoint is that nothing compels us to put 4 after 2+2=..., just as nothing forces Lewis Carroll's tortoise to infer Q from "P, and if P, then Q". Calculating is a practice, it involves the application and mastery of a technique, and nothing to do with discovering mythological objects in the heavens. But, surely our practices might have been different; how then, is the hardness of the logical must to be accounted for? The Wittgensteinian response, in a sense, is to reconstruct our misleading notions of necessity along naturalistic, Quinean lines. It is best to lift it straight from Wittgenstein I think:

""But mathematical truth is independent of whether human beings know it or not!"—Certainly, the propositions "Human beings believe that twice two is four" and "Twice two is four" do not mean the same. The latter is a mathematical proposition; the other, if it makes sense at all, may perhaps mean: human beings have arrived at the mathematical proposition. The two propositions have entirely different uses.—But what would this mean: "Even though everybody believed that twice two was five it would still be four"?—For what would it be like for everybody to believe that?—Well, I could imagine, for instance, that people had a different calculus, or a technique which we should not call "calculating". But would it be wrong? (Is a coronation wrong? To beings different from ourselves it might look extremely odd.)"

In other words, certainly, our practices might have been different; we might have constructed for ourselves a completely different form of life into which the notions of numbers and mathematics have no place or application. This, however, properly understood, is simply to say that we would have had different practices, practices different from our actual practices of calculating. By way of an analagy, might the rules of chess have been different? Well, certainly, we might have played a different game.

I, personally, take a slightly more liberal view than the reading of Wittgenstein suggested above (although, it is more than plausible that he agrees with me). I would say, that in certain circumstances, it would be right to call different practices from our actual ones "calculating". Clearly, were people to have a different calculus such that 2+2=5 and a whole raft of other results, they would have to have different practices that related to these; for, if they simply said that 2+2=5 and went on in exactly the same way we do, we should simply say they were speaking a different language, but had the same mathematics. However, it is conceivable that these different practices might be very similar to ours, for example, if their calculus was still used to build bridges, measure distances, and derive theoretical results. Clearly, the mathematical techniques would have to be applied, and would result in different consequences, but with sufficient similarity to our actual practices, there seems to be no problem with saying, "They calculate differently to us." Imagine, for example, we made contact with a tribe that played a board game identical to our own chess, except without the bishops. Do they have a radically different practice, or do they merely have a "different form of chess"?

Another important point against the platonist account of mathematical necessity, hinted at in Wittgenstein's words above, is that it leaves open the possibility that we have always been wrong in our belief, for example, that 2+2=4. The platonist says that mathematical entities stand in an eternal relation to one another in the platonic heavens, independently of whatever we believe of them, thus, we may one day discover that all of human kind has always been wrong about 2+2=4, and that in fact, this relation does not hold. This seems utterly absurd to me.

Quote:

I know what you mean, and you are simply wrong. Abstract objects and their properties are posited by platonist. The AO 3 has the property of being prime, and so '3 is prime' is true. If you doubt AO exist, then you simply don` t know AO is a posit. If you doubt AO 3 has the property of prime, then you simply don ` t know the property of AO 3.

You are espousing a form of constructivism, here. Platonists do not posit anything, platonists say that mathematical objects exist, independently of human mental states. Platonism has become practically synonymous with realism, clearly, if something is merely "posited", it is not real.

Quote:

The justification for platonism is quite obvious. Platonism explains all the facts we want in a theory that explains the objectivity of mathematics. It is also the best view compare to all the rivaling implausible views.

As I have quite clearly demonstrated throughout this thread, platonism offers no explanations. Its explanation of necessity is that objects stand in eternal relations to one another; its explanation of the subject matter of mathematics is abstract objects that actually exist. These are not answers, they are mystical babblings. What is an eternal relation? What is an abstract object? Where do they they exist? How do I know about them? How do my words refer to them? Platonism explains nothing. Moreover, the idea of shadowy objects that somehow necessarily exist, in a "place" without spacio-temporal properties, that we apparently know about but we do not know how we know about, is not merely implausible, it is nonsensical.


Quote:
Well, there is a famous paper in the field that said one can look at the integers as sets. The author show there are many ways to construct these sets. This is why the author of "platonism, and anti platonism" said there need not be any uniqueness. This is consistent with '3 is prime' corresponds to only one AO 3.
Quote:
On closer reading, I think perhaps Emil was talking about the indeterminacy of reference. One argument in favour of this: On one model, M, we might map "3 is prime" to a an abstract object, AO1, and map "3 is odd" to AO2. Imagine we now reverse the mapping so that we map "3 is prime" to AO, and map "3 is odd" to AO1, and call this M*. Now, in M and M* both "3 is prime" and "3 is odd" is true, but they represent the platonic heavens differently, which one is right? All of our observations and behaviour are compatible with either.


This is not a problem as i said before.

So now you are saying that numbers are sets? Presumably, the only way that this view can be consistent with platonism is if sets are themselves platonic, or you are saying that the sets express integers, but refer to platonic entities. Now, depending on how you define 0 and the the successor function, 2 can be constructed as, {{ }, {{ }}}, or with other definitions, it could be constructed as {{{ }}}. My knowledge of set theory is pretty rudimentary, but as I understand it these are two different sets (correct?). If you are committed to the view that numbers are sets and that sets are platonic entites, then you are committed to the view that there is not a number two, but there are in fact two number twos. Sometimes we as the first number two to the second number two to get one of the many number fours? In essence, you would be committed to saying that '2+2=4' can be used to express a number (presumably an infinity) of different true propositions. Clearly, this is absurd, and in any case, you still have the problem of how the signs {{ }, {{ }}} and {{{ }}} refer/map on to the platonic sets.

Likewise, if you are saying that these different constructions all refer to the same platonic entity, 2, the problem of indeterminacy of reference remains. All of our observations, behaviour and results are compatible with a number of different ways of mapping our signs onto mathematical reality. Platonism tells us that there is only one correct mapping, yet there is no fact of the matter that determines which mapping is correct.




You say a whole lot of useless stuff i am not at all interested to comment. Let ` s talk about objectivity, since it is here that platonism explains best of the semantics of math statements. Like i said, language games notion do not capture the notion of "truth in all possible worlds" view of math. How ever you called 'practices', or 'language games', it would not solve this problem. sorry. Also, saying math is like chess would not work, since formalism in incompatible with Godel theorem.

You comment about how platonism tell us math objects just exist, and not a posit is rather stupid. To "posit" something is like saying it exist. There are different, but former do imply the latter.


You constantly say Platonism is mystical. What a waste of words. Perhaps it is mystical because you just don` t get it. In the past, people thought action -at -a- distance was mystical. It is not mystical anymore.

I think it is pretty clear unique references is a difficult problem. It is not relevant to us at all. Platonist don ` t to commit to unique reference of AO, or explication the relationships between AO in plato` s haven.





ughaibu
 
  1  
Reply Sat 10 Jul, 2010 09:23 pm
@TuringEquivalent,
TuringEquivalent wrote:
ughaibu wrote:
generalisation
the concrete object, cat
Even without a philosophical and mathematical training, you should be able to appreciate that a particular is not the general.
TuringEquivalent
 
  1  
Reply Sat 10 Jul, 2010 10:00 pm
@ughaibu,
ughaibu wrote:

TuringEquivalent wrote:
ughaibu wrote:
generalisation
the concrete object, cat
Even without a philosophical and mathematical training, you should be able to appreciate that a particular is not the general.



So, a particular is not general( universal? ). Is this why you think concrete objects are AO for you? I am not trying to insult, but you are making very simple mistakes. I feel you are not being intellectually honest at all.


ughaibu
 
  1  
Reply Sat 10 Jul, 2010 10:01 pm
@TuringEquivalent,
TuringEquivalent wrote:
Is this why you think concrete objects are AO ?
What on Earth are you talking about?
TuringEquivalent
 
  1  
Reply Sat 10 Jul, 2010 10:09 pm
@ughaibu,
ughaibu wrote:

TuringEquivalent wrote:
Is this why you think concrete objects are AO ?
What on Earth are you talking about?



You made this statement:

Quote:
if abstract objects are required for mathematical statements to be true,then they're required for any generalisation to be true.


I pointed out how absurd it is above. The "they" refers to AOs. "Any generalization" would include statements like 'cat is in the mat'. Surely, cat is not an AO, unless you think concrete objects are the same as AO.

ughaibu
 
  1  
Reply Sat 10 Jul, 2010 10:12 pm
@TuringEquivalent,
TuringEquivalent wrote:
"Any generalization" would include statements like 'cat is in the mat'.
Of course it wouldn't!
mickalos
 
  2  
Reply Sat 10 Jul, 2010 10:52 pm
@TuringEquivalent,
TuringEquivalent wrote:

You say a whole lot of useless stuff i am not at all interested to comment.

This response is not only rude, but it shows a complete lack of intellectual character. When you say things that I believe to be wrong, irrelevant, or "useless", I outline in detail why I think so. You are simply dismissive, despite the fact that my comments are quite clearly relevant to the philosophy of mathematics. Indeed, in taking the Wittgensteinian view, I am essentially parroting what Wittgenstein says in his six volume Remarks on the Foundations of Mathematics. You cannot simply dismiss this as "useless" without justification.

Quote:
Let ` s talk about objectivity, since it is here that platonism explains best of the semantics of math statements. Like i said, language games notion do not capture the notion of "truth in all possible worlds" view of math. How ever you called 'practices', or 'language games', it would not solve this problem. sorry.


The vast majority of my post is about necessity (you seem to be using objectivity in place of necessity), yet you refuse to engage with it at all. As I said above, the Wittgensteinian view does not try to capture a "true in all possible worlds view of math" it attempts to reconstruct the traditional notion of necessity. You have refused to address my points and defend yourself so often now that I hardly need to reformulate them, and thus I shan't. Perhaps you would like to read this more carefully:

You are talking about necessity rather than objectivity. It should be clear that necessity is an even bigger problem for the platonist. Platonists construe the subject matter of mathematics as objects of some description; how on earth are we to understand the notion that any given platonic entity that corresponds to a number necessarily exists, and necessarily has certain properties? This clearly brings out why modelling mathematical facts on descriptive facts creates a confused chimera of a theory. The notion of a table having necessary existence does not make sense, nor does it make sense that a ball could be necessarily red. Yet the Platonist claims that this is exactly the case with abstract objects, that somehow this logical necessity exists over and above our practices, but offers only mysticism as an explanation. In short, platonism does not offer an account of logical necessity.

As we have already discussed, the Wittgensteinian viewpoint is that nothing compels us to put 4 after 2+2=..., just as nothing forces Lewis Carroll's tortoise to infer Q from "P, and if P, then Q". Calculating is a practice, it involves the application and mastery of a technique, and nothing to do with discovering mythological objects in the heavens. But, surely our practices might have been different; how then, is the hardness of the logical must to be accounted for? The Wittgensteinian response, in a sense, is to reconstruct our misleading notions of necessity along naturalistic, Quinean lines. It is best to lift it straight from Wittgenstein I think:

""But mathematical truth is independent of whether human beings know it or not!"—Certainly, the propositions "Human beings believe that twice two is four" and "Twice two is four" do not mean the same. The latter is a mathematical proposition; the other, if it makes sense at all, may perhaps mean: human beings have arrived at the mathematical proposition. The two propositions have entirely different uses.—But what would this mean: "Even though everybody believed that twice two was five it would still be four"?—For what would it be like for everybody to believe that?—Well, I could imagine, for instance, that people had a different calculus, or a technique which we should not call "calculating". But would it be wrong? (Is a coronation wrong? To beings different from ourselves it might look extremely odd.)"

In other words, certainly, our practices might have been different; we might have constructed for ourselves a completely different form of life into which the notions of numbers and mathematics have no place or application. This, however, properly understood, is simply to say that we would have had different practices, practices different from our actual practices of calculating. By way of an analagy, might the rules of chess have been different? Well, certainly, we might have played a different game.

I, personally, take a slightly more liberal view than the reading of Wittgenstein suggested above (although, it is more than plausible that he agrees with me). I would say, that in certain circumstances, it would be right to call different practices from our actual ones "calculating". Clearly, were people to have a different calculus such that 2+2=5 and a whole raft of other results, they would have to have different practices that related to these; for, if they simply said that 2+2=5 and went on in exactly the same way we do, we should simply say they were speaking a different language, but had the same mathematics. However, it is conceivable that these different practices might be very similar to ours, for example, if their calculus was still used to build bridges, measure distances, and derive theoretical results. Clearly, the mathematical techniques would have to be applied, and would result in different consequences, but with sufficient similarity to our actual practices, there seems to be no problem with saying, "They calculate differently to us." Imagine, for example, we made contact with a tribe that played a board game identical to our own chess, except without the bishops. Do they have a radically different practice, or do they merely have a "different form of chess"?

Another important point against the platonist account of mathematical necessity, hinted at in Wittgenstein's words above, is that it leaves open the possibility that we have always been wrong in our belief, for example, that 2+2=4. The platonist says that mathematical entities stand in an eternal relation to one another in the platonic heavens, independently of whatever we believe of them, thus, we may one day discover that all of human kind has always been wrong about 2+2=4, and that in fact, this relation does not hold. This seems utterly absurd to me.

Quote:
Also, saying math is like chess would not work, since formalism in incompatible with Godel theorem.

Wittgenstein's attack on the platonic account of mathematical inference rests on a more general attack of a platonic idea of the inference rules of any formal system. He is arguing against formalism just as much as he is arguing against platonism, for formalism rests the idea that the supposedly abstract rules of logic inexorably determine an application. As I have pointed out above, Wittgenstein does not think that there is any fact of the matter, syntactic, mental, abstract, or otherwise, that we can infer Q from "P, and if P, then Q". Indeed, Wittgenstein does not just object to platonic accounts of formal rules of inference, but of all rules, including the rules of chess.

Quote:
You comment about how platonism tell us math objects just exist, and not a posit is rather stupid. To "posit" something is like saying it exist. There are different, but former do imply the latter.

Yes, when one posits an object it is like saying something exists; one assumes that an object exists, and then talks about it as if it does, regardless of whether or not it actually does. Clearly, this is not the same thing as an object actually existing. One is counterfactual, the other is actual. The first line of the Stanford encyclopaedia of philosophy article on platonism in the philosophy of mathematics is "Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices." Now, the world does not posit things, positing things is something that people do. Get the picture?

Quote:

You constantly say Platonism is mystical. What a waste of words. Perhaps it is mystical because you just don` t get it. In the past, people thought action -at -a- distance was mystical. It is not mystical anymore.

Yes, it is no longer mystical because scientists have explained it adequately for it to fit in nicely with my other beliefs. However, I have not heard any explanation of how anything can be said to necessarily exist, and moreover, exist without spacio-temporal properties, that fits in with generally accepted beliefs about the world and the things in it. You have not even tried.

Quote:
I think it is pretty clear unique references is a difficult problem. It is not relevant to us at all. Platonist don ` t to commit to unique reference of AO, or explication the relationships between AO in plato` s haven.

Firstly, on your view, there is only one platonic heaven, and the objects in it are related to each other in a determinate way. If a mathematical proposition is true iff it correctly corresponds to platonic objects and the relations between them, then in order to say something true about a number you must pick out that number with your words and say something determinate and accurate about its properties and the relation it stands in to other objects. If you cannot uniquely refer to an object then we cannot pick them out with our words, and if Putnam's model-theoretic argument is correct, which I vaguely outlined above, then what we say and what we say them about is under determined. If platonism cannot account for these problems, the consequence is that if platonism were true, we would not even be able to talk about mathematics.

Clearly this is not a problem that you can be nonchalant about if you are going to offer even so much as the pretence of intellectual integrity.

Quote:
"Any generalization" would include statements like 'cat is in the mat'.

A general statement would be, "All cats are on the mat." More formally:
For all x there exists a y such that ((if (x is a cat), then (x is on y)) & (there is no more than one y))
TuringEquivalent
 
  1  
Reply Sun 11 Jul, 2010 04:20 pm
@ughaibu,
ughaibu wrote:

TuringEquivalent wrote:
"Any generalization" would include statements like 'cat is in the mat'.
Of course it wouldn't!


'any' stands for the universal quantifier 'for all'. I know you enough that you obvious know symbolic logic, so, why would you make these stupid mistakes?
0 Replies
 
TuringEquivalent
 
  1  
Reply Sun 11 Jul, 2010 05:58 pm
@mickalos,
look, i am not here to write a lengthy paper. My time is better spend in a more productive way, so if you write, long posts, i am going to abstract, and simplify to some main point. That said, I will respond in parts when i have time.
 

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