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Best argument for Platonism.

 
 
Reply Thu 8 Jul, 2010 08:34 am
The best argument for mathematical Platonism has to be the fact that other views are so implausible. Intuitionism is really like the Kantian view, and it is the view that mathematical objects are mental objects. This view sucks, because it cannot account for the objectivity of math. Formalism does work, because of Godel theorem. Fictionism say that all the mathematical statements we made are false, but that is rather counterintuitive. Structuralism say that math is really structures, de re, but that looks a lot like Platonism.
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kennethamy
 
  1  
Reply Thu 8 Jul, 2010 03:02 pm
@TuringEquivalent,
TuringEquivalent wrote:

The best argument for mathematical Platonism has to be the fact that other views are so implausible. Intuitionism is really like the Kantian view, and it is the view that mathematical objects are mental objects. This view sucks, because it cannot account for the objectivity of math. Formalism does work, because of Godel theorem. Fictionism say that all the mathematical statements we made are false, but that is rather counterintuitive. Structuralism say that math is really structures, de re, but that looks a lot like Platonism.


Your argument commits the ad ignorantium fallacy. And, of course, that is independent of whether your premise is true.
TuringEquivalent
 
  1  
Reply Thu 8 Jul, 2010 03:39 pm
@kennethamy,
kennethamy wrote:

TuringEquivalent wrote:

The best argument for mathematical Platonism has to be the fact that other views are so implausible. Intuitionism is really like the Kantian view, and it is the view that mathematical objects are mental objects. This view sucks, because it cannot account for the objectivity of math. Formalism does work, because of Godel theorem. Fictionism say that all the mathematical statements we made are false, but that is rather counterintuitive. Structuralism say that math is really structures, de re, but that looks a lot like Platonism.


Your argument commits the ad ignorantium fallacy. And, of course, that is independent of whether your premise is true.


It applies to you. An example of that fallacy is:

"In spite of all the talk, not a single flying saucer report has been authenticated. We may assume, therefore, there are not such things as flying saucers."

You love to jump from something being unlikely, and implausible to it being false.

0 Replies
 
mickalos
 
  1  
Reply Thu 8 Jul, 2010 05:14 pm
@kennethamy,
kennethamy wrote:

TuringEquivalent wrote:

The best argument for mathematical Platonism has to be the fact that other views are so implausible. Intuitionism is really like the Kantian view, and it is the view that mathematical objects are mental objects. This view sucks, because it cannot account for the objectivity of math. Formalism does work, because of Godel theorem. Fictionism say that all the mathematical statements we made are false, but that is rather counterintuitive. Structuralism say that math is really structures, de re, but that looks a lot like Platonism.


Your argument commits the ad ignorantium fallacy. And, of course, that is independent of whether your premise is true.

If the argument has the form: PvQvRvI, ~Q, ~R, ~I ¦- P
Then the argument isn't structurally poor. I think the main flaws are, firstly, platonism, intutionism, structuralism and formalism do not exhaust all the possibilities. More fundamentally, you cannot invoke implausibility and counter intuitiveness in favour of platonism! What on earth is the platonic form of the two? Call it an abstract object if you like, but you are still talking nonsense. I understand what an abstract painting is, but I have no idea what an abstract object is supposed to be: Do they exist? Apparently. Where do they exist? In the realm of abstract objects. Where is that? ... They're making it up as they go along!

I think that there are mathematical facts; that is to say, I think that 2+2=4 is true. However, it seems perfectly obvious to me that it is a gross error, a kind of category mistake, I suppose, to model mathematical facts on empirical facts. Why would anybody think that 2+2=4 is true in the same way that "The cat is on the mat" is true?

I would plumb for a Wittgensteinian conventionalism about mathematics. Does anything in "2+2=" compel us to answer 4? Surely nothing in the symbols, they are just symbols. Anything in my mind? I don't believe I need be thinking anything when I answer 4, it is instinctive; certainly, sometimes when I am asked to continue a series, or solve a problem that has caused me trouble for days, I don't have to do any thinking at all, sometimes I just see the answer, and all of a sudden I know how to go on. Is there anything in the rule for addition that forces an application? Under some interpretation I can make any action accord with it. Is there something in reality that forces 2+2=4 upon us? A platonic form? Show me this bit of reality!

Calculating is a practice, and it is constitutive of that practice that 2+2=4. What makes it so that 2+2=4? That is like asking what makes it so that the pawns can move forward two squares the first time they are moved in a game of chess. What are you asking for? A justification of some kind? Based on reason? Based on experience? You will get none, that is just how we play chess.
TuringEquivalent
 
  0  
Reply Thu 8 Jul, 2010 05:37 pm
@mickalos,
mickalos wrote:

kennethamy wrote:

TuringEquivalent wrote:

The best argument for mathematical Platonism has to be the fact that other views are so implausible. Intuitionism is really like the Kantian view, and it is the view that mathematical objects are mental objects. This view sucks, because it cannot account for the objectivity of math. Formalism does work, because of Godel theorem. Fictionism say that all the mathematical statements we made are false, but that is rather counterintuitive. Structuralism say that math is really structures, de re, but that looks a lot like Platonism.


Your argument commits the ad ignorantium fallacy. And, of course, that is independent of whether your premise is true.

If the argument has the form: PvQvRvI, ~Q, ~R, ~I ¦- P
Then the argument isn't structurally poor. I think the main flaws are, firstly, platonism, intutionism, structuralism and formalism do not exhaust all the possibilities. More fundamentally, you cannot invoke implausibility and counter intuitiveness in favour of platonism! What on earth is the platonic form of the two? Call it an abstract object if you like, but you are still talking nonsense. I understand what an abstract painting is, but I have no idea what an abstract object is supposed to be: Do they exist? Apparently. Where do they exist? In the realm of abstract objects. Where is that? ... They're making it up as they go along!

I think that there are mathematical facts; that is to say, I think that 2+2=4 is true. However, it seems perfectly obvious to me that it is a gross error, a kind of category mistake, I suppose, to model mathematical facts on empirical facts. Why would anybody think that 2+2=4 is true in the same way that "The cat is on the mat" is true?

I would plumb for a Wittgensteinian conventionalism about mathematics. Does anything in "2+2=" compel us to answer 4? Surely nothing in the symbols, they are just symbols. Anything in my mind? I don't believe I need be thinking anything when I answer 4, it is instinctive; certainly, sometimes when I am asked to continue a series, or solve a problem that has caused me trouble for days, I don't have to do any thinking at all, sometimes I just see the answer, and all of a sudden I know how to go on. Is there anything in the rule for addition that forces an application? Under some interpretation I can make any action accord with it. Is there something in reality that forces 2+2=4 upon us? A platonic form? Show me this bit of reality!

Calculating is a practice, and it is constitutive of that practice that 2+2=4. What makes it so that 2+2=4? That is like asking what makes it so that the pawns can move forward two squares the first time they are moved in a game of chess. What are you asking for? A justification of some kind? Based on reason? Based on experience? You will get none, that is just how we play chess.


What a waste of words. Your whole point is that mathematical truth is grounded on mathematical practices. This would not explain the objectivity of mathematics. If it is just culture, then why the same mathematical theorem are discovery in different cultures? Why don` t the theorems contradict one another? Godel ` s theorem is not going away if you think these math conventions are rules.
ughaibu
 
  1  
Reply Thu 8 Jul, 2010 06:16 pm
@TuringEquivalent,
TuringEquivalent wrote:
Your whole point is that mathematical truth is grounded on mathematical practices.
I thought his main point was that the competition cant be eliminated on vague pretexts which apply at least equally to Platonism itself.
mickalos
 
  1  
Reply Thu 8 Jul, 2010 06:48 pm
@TuringEquivalent,
TuringEquivalent wrote:


What a waste of words. Your whole point is that mathematical truth is grounded on mathematical practices. This would not explain the objectivity of mathematics. If it is just culture, then why the same mathematical theorem are discovery in different cultures? Why don` t the theorems contradict one another? Godel ` s theorem is not going away if you think these math conventions are rules.


I would hardly call it a waste of words. The points against Platonism's queerness are perfectly valid, and are, I should think, the primary reasons why it is so out of favour nowadays.

Indeed, my whole point is that mathematical truth is grounded on our practices of calculating, proving, inferring etc. What makes one theorem follow from another? You seem think it is some mysterious fact about reality, and yet you cannot point to it, you cannot say how this particular fact makes it so that 2+2=4, nor can you even describe this occult fact in anything other than meaningless terms such as "abstract object" or "platonic entity". Presumably you want to say that something corresponds to 2+2=4. Is this something the same sort of thing as a dog or a rock? Clearly not. Then what is it? How does it correspond? You have provided no answers to these questions, and I cannot think of any satisfactory account that has been given.

How, then, do I justify 2+2=4? Reason will not work, for every justification stands in need of a justification itself, and indeed, in need of a justification to be treated as a justification. Calculations, like the moves in a game of chess, are constitutive of a particular practice, they do not stand in need of a justification. To quote Wittgenstein, "If I have exhausted the justifications I have reached bedrock, and my spade is turned. Then I am inclined to say: "This is simply what I do.""

If you think Wittgensteinian conventionalism, or what I said above entails some form of cultural relativism, I think you are quite clearly wrong. Mathematics is not a cultural practice, but a human practice, or to use the Wittgenstein's terminology, a human form of life. The Greeks counted in just the same way you and I do; at least, certain Greek practices can be very easily identified with our own. The Russians play the same game when they sit down at a chessboard as the Indians do. Cultures have their particular practices, but so does humanity at large. I would have thought the conventionalist explanation of objectivity is obvious: No deviations are permitted. When a child says, "2+2=5", his individuality is not celebrated, instead we demand that he judges in conformity with the rest of mankind. Perhaps conventionalism may entail some kind of conceptual relativism, but I think the idea of conceptual relativism may be somewhat empty; less interesting and threatening that it at first sounds, Davidson's "On the very idea of a conceptual scheme" is a good analysis of this. It certainly compromises the hardness of the logical must, but I do not think this is such a bad thing; the idea of logical compulsion is, while perfectly understandable on a psychological level, metaphysically very strange.


TuringEquivalent
 
  1  
Reply Thu 8 Jul, 2010 09:12 pm
@ughaibu,
ughaibu wrote:

TuringEquivalent wrote:
Your whole point is that mathematical truth is grounded on mathematical practices.
I thought his main point was that the competition cant be eliminated on vague pretexts which apply at least equally to Platonism itself.


What is vague? It is very clear to me. There are not that many views, and all those views either are so implausible, and thus worthless, or they reduced to one of the listed views in the op post.
ughaibu
 
  1  
Reply Thu 8 Jul, 2010 09:43 pm
@TuringEquivalent,
TuringEquivalent wrote:
What is vague? It is very clear to me. There are not that many views, and all those views either are so implausible, and thus worthless, or they reduced to one of the listed views in the op post.
Implausibility and counter-intuitiveness are both vague, and Platonism suffers from both.
0 Replies
 
TuringEquivalent
 
  1  
Reply Thu 8 Jul, 2010 09:48 pm
@mickalos,
mickalos wrote:

TuringEquivalent wrote:


What a waste of words. Your whole point is that mathematical truth is grounded on mathematical practices. This would not explain the objectivity of mathematics. If it is just culture, then why the same mathematical theorem are discovery in different cultures? Why don` t the theorems contradict one another? Godel ` s theorem is not going away if you think these math conventions are rules.


I would hardly call it a waste of words. The points against Platonism's queerness are perfectly valid, and are, I should think, the primary reasons why it is so out of favour nowadays.

Indeed, my whole point is that mathematical truth is grounded on our practices of calculating, proving, inferring etc. What makes one theorem follow from another? You seem think it is some mysterious fact about reality, and yet you cannot point to it, you cannot say how this particular fact makes it so that 2+2=4, nor can you even describe this occult fact in anything other than meaningless terms such as "abstract object" or "platonic entity". Presumably you want to say that something corresponds to 2+2=4. Is this something the same sort of thing as a dog or a rock? Clearly not. Then what is it? How does it correspond? You have provided no answers to these questions, and I cannot think of any satisfactory account that has been given.

How, then, do I justify 2+2=4? Reason will not work, for every justification stands in need of a justification itself, and indeed, in need of a justification to be treated as a justification. Calculations, like the moves in a game of chess, are constitutive of a particular practice, they do not stand in need of a justification. To quote Wittgenstein, "If I have exhausted the justifications I have reached bedrock, and my spade is turned. Then I am inclined to say: "This is simply what I do.""

If you think Wittgensteinian conventionalism, or what I said above entails some form of cultural relativism, I think you are quite clearly wrong. Mathematics is not a cultural practice, but a human practice, or to use the Wittgenstein's terminology, a human form of life. The Greeks counted in just the same way you and I do; at least, certain Greek practices can be very easily identified with our own. The Russians play the same game when they sit down at a chessboard as the Indians do. Cultures have their particular practices, but so does humanity at large. I would have thought the conventionalist explanation of objectivity is obvious: No deviations are permitted. When a child says, "2+2=5", his individuality is not celebrated, instead we demand that he judges in conformity with the rest of mankind. Perhaps conventionalism may entail some kind of conceptual relativism, but I think the idea of conceptual relativism may be somewhat empty; less interesting and threatening that it at first sounds, Davidson's "On the very idea of a conceptual scheme" is a good analysis of this. It certainly compromises the hardness of the logical must, but I do not think this is such a bad thing; the idea of logical compulsion is, while perfectly understandable on a psychological level, metaphysically very strange.





Platonism is out of favor? Are you joking me? It is the majority view in academic philosophy. The only rival view is fictionalism, and that view is counterintuitive. For the second half of the last century, there emerge an array of metaphysical theories.

I guess your problem with platonism is that you just don` t like abstract objects. Well, that is your bias. Some people just don` t like metaphysics, but there is nothing wrong with platonism as an explanatory framework. The view itself is rather counterintuitive from our every experience, since most of knowledge have a casual component. How do you know the ball is there? Because you can kick it. What is the justification that all knowledge must be this way? Is there any reason?

Quote:

No deviations are permitted. When a child says, "2+2=5", his individuality is not celebrated, instead we demand that he judges in conformity with the rest of mankind. Perhaps conventionalism may entail some kind of conceptual relativism, but I think the idea of conceptual relativism may be somewhat empty; less interesting and threatening that it at first sounds, Davidson's "On the very idea of a conceptual scheme" is a good analysis of this. It certainly compromises the hardness of the logical must, but I do not think this is such a bad thing; the idea of logical compulsion is, while perfectly understandable on a psychological level, metaphysically very strange
.

This makes no sense. Mathematical statements cannot be agreed on by the majority. At present, there is no consensus that all even numbers greater 2 is the sum of two primes, but conceivably, the answer is either true, or false.









ughaibu
 
  1  
Reply Thu 8 Jul, 2010 09:57 pm
@TuringEquivalent,
TuringEquivalent wrote:
there is nothing wrong with platonism as an explanatory framework. The view itself is rather counterintuitive from our every experience, since most of knowledge have a casual component.
1) Platonism doesn't manage to explain anything
2) it's not just counter-intuitive, the defining features of abstract objects are the same features that define the non-existence of other objects.
TuringEquivalent
 
  1  
Reply Thu 8 Jul, 2010 10:34 pm
@ughaibu,
ughaibu wrote:

TuringEquivalent wrote:
there is nothing wrong with platonism as an explanatory framework. The view itself is rather counterintuitive from our every experience, since most of knowledge have a casual component.
1) Platonism doesn't manage to explain anything
2) it's not just counter-intuitive, the defining features of abstract objects are the same features that define the non-existence of other objects.


1
Platonism explains the objectivity of mathematics in the same way that we can all agree there is a table in front of us by point at it.

2. This is ridiculous. Abstract objects are "posited" objects to help explain the objectivity of mathematics. If "something" is non existent, why are there properties?

ughaibu
 
  1  
Reply Thu 8 Jul, 2010 10:50 pm
@TuringEquivalent,
TuringEquivalent wrote:
1
Platonism explains the objectivity of mathematics in the same way that we can all agree there is a table in front of us by point at it.
I'm not convinced that there is any objectivity, of mathematics, that can be isolated from the formalism. You claim that the same theorems are proved within distinct cultures, but this ignores the fact that, beyond naive geometry and numerical calculation, mathematics consists of various cultures, and across different forms of mathematics, incompatible theorems are proved.
TuringEquivalent wrote:
2. This is ridiculous. Abstract objects are "posited" objects to help explain the objectivity of mathematics. If "something" is non existent, why are there properties?
Gods are objects posited to help explain various phenomena, I very much doubt that you consider this to be any kind of reason to adopt realism about gods, so I equally expect you to find your own argument unconvincing, I'm certainly unconvinced. On the question of properties, "where" are the properties of fictional objects?

According to Emil, there is an, at least countable, infinity of abstract objects corresponding to the proposition 1+1=x and xNot Equal2, do you hold this position, or do you hold that there is only one abstract object of the form 1+1=x and that abstract object corresponds to the proposition 1+1=2?
mickalos
 
  1  
Reply Fri 9 Jul, 2010 05:20 am
@TuringEquivalent,
TuringEquivalent wrote:

Platonism is out of favor? Are you joking me? It is the majority view in academic philosophy. The only rival view is fictionalism, and that view is counterintuitive. For the second half of the last century, there emerge an array of metaphysical theories.

As a philosophy undergraduate currently applying to graduate courses, you may have been correct about the dominance of platonism fifty years ago, but I assure you that this is no longer the case. The objections to platonism have mounted up, and philosophers today are more open to alternatives to the correspondence theory of truth, especially in areas like mathematics and ethics. There are a plethora of alternatives to platonism, from naturalism to quasi-realism, to suggest that platonism is correct because fictionalism is counterintuitive (though I find fictionalism no more counter intuitive than platonism) is absurd.

Quote:
I guess your problem with platonism is that you just don` t like abstract objects. Well, that is your bias. Some people just don` t like metaphysics, but there is nothing wrong with platonism as an explanatory framework. The view itself is rather counterintuitive from our every experience, since most of knowledge have a casual component. How do you know the ball is there? Because you can kick it. What is the justification that all knowledge must be this way? Is there any reason?

You make it sound as if I object to platonism on a mere whim. I object to the idea of abstract objects because the notion of a non-causal, non-contingent, non-spacio-temporal object does not make sense. In what sense can an object be said to exist, and yet have no mass, no point of creation, and no causal connection with the world or anything in it? You are also quite right to say that there is an epistemic problem as well. How can the platonist justify his knowledge that 2+2=4 when he cannot explain his relation to the objects that he posits? Indeed, as Ughaibu points out, it does not seem as if the platonist can even individuate his immaterial objects (a point also made by Strawson against substance dualism). Indeed, if abstract objects are not causally hooked up with the world, and if one cannot individuate platonic entities with a definite description in a way that does not incur some form of circularity, then it is unclear how we can even refer to platonic entities.

I do not think I ever claimed that all knowledge must come to us via some kind of truth preserving causal link. Indeed, I do not think the Wittgensteinian view that I take could be said to suggest that there is a truth preserving causal link between my knowledge that 2+2=4 and the "fact" that 2+2=4; certainly not in the way that people who adhere to causal theories of knowledge want to say, in any case. However, I do not think that it is enough to say "this is not the only possible way", if you are to justify your knowledge claims you must give an account of how you know, not just enumerate the ways in which you do not know. You have failed to do this. You have not even attempted to do this, which suggests you do not even know where to begin.

Quote:

This makes no sense. Mathematical statements cannot be agreed on by the majority. At present, there is no consensus that all even numbers greater 2 is the sum of two primes, but conceivably, the answer is either true, or false.

That 2+2=4 is true is not a matter of tallying up the number of people who agree with it against those who don't. 2+2=4 is not an empirical hypothesis, it is a rule, and as such anybody's reasons for disagreeing with it will be discounted. You are quite right, the Goldbach conjecture is something that is treated as either true or false, but my point (and Wittgenstein's) is that no proof or "fact" about the world (because mathematical facts simply are not the same as physical facts) can force me to come down on a particular side. The tracks are not already laid out into the darkness, we may extend them in any direction we like; one might as well say that chess only had to be discovered, it was always there, even before anybody had played it, or even thought of playing it. In many ways, like the first people to play chess, mathematicians are inventors, not discoverers.

Quote:
1
Platonism explains the objectivity of mathematics in the same way that we can all agree there is a table in front of us by point at it.

We cannot point to 2+2=4.


Quote:
2. This is ridiculous. Abstract objects are "posited" objects to help explain the objectivity of mathematics. If "something" is non existent, why are there properties?

Now it sounds like you are arguing for fictionalism. I can posit all sorts of objects in all sorts of posited realms; I posit that you are moonwalking through a Baudrillardian hyperspace wearing a tutu, that doesn't make it true, nor does it bring into existence any platonic realm. If mathematics merely make-believe, or posited, then it is no more true than a fairy tail. This is not what platonists argue. Platonism states that mathematical objects actually exist, and they are causally independent, and mind independent of anything we may do or posit.
kennethamy
 
  1  
Reply Fri 9 Jul, 2010 07:30 am
The Platonic view that fire-engines are not red, but that only Redness is red, is really too goofy to take seriously, and that epitomizes Platonism.
ughaibu
 
  1  
Reply Fri 9 Jul, 2010 08:22 am
@kennethamy,
kennethamy wrote:
The Platonic view that fire-engines are not red, but that only Redness is red, is really too goofy to take seriously, and that epitomizes Platonism.
Yet you appear to be a realist about abstract objects.
0 Replies
 
TuringEquivalent
 
  1  
Reply Fri 9 Jul, 2010 08:40 am
@ughaibu,
ughaibu wrote:

TuringEquivalent wrote:
1
Platonism explains the objectivity of mathematics in the same way that we can all agree there is a table in front of us by point at it.
I'm not convinced that there is any objectivity, of mathematics, that can be isolated from the formalism. You claim that the same theorems are proved within distinct cultures, but this ignores the fact that, beyond naive geometry and numerical calculation, mathematics consists of various cultures, and across different forms of mathematics, incompatible theorems are proved



I am not sure formalism captures mathematical truths. Godel show that there will be always arithmetic facts that are not deducible within the system.

Also, there is no difficulty for different branches of math, since different areas of mathematics simply describes different classes of abstract objects in the platonic world view. So, Euclid geometry describes one classes of objects, and spherical geometry describes another.

Quote:
Gods are objects posited to help explain various phenomena, I very much doubt that you consider this to be any kind of reason to adopt realism about gods, so I equally expect you to find your own argument unconvincing, I'm certainly unconvinced. On the question of properties, "where" are the properties of fictional objects?



I don ` t posit God, but i do posit nomic necessity, or dispositional properties of natural kind. This is technical, but to be brief, i believe in laws of nature.

I don ` t want to get into fictional objects. Lets stick to platonic ones.

Quote:

According to Emil, there is an, at least countable, infinity of abstract objects corresponding to the proposition 1+1=x and xNot Equal2, do you hold this position, or do you hold that there is only one abstract object of the form 1+1=x and that abstract object corresponds to the proposition 1+1=2?


I am not familiar with the technical details of Emil claim, but i don` t see any problem with why a mathematical proposition cannot map into many abstract objects. The details are not relevant. In the book "platonism, and anti-platonism", the author say there need not be "uniqueness" to the mapping at all. It is consistent to maintain platonism, even tho there is not unique correspondence between math statements, and abstract objects.
TuringEquivalent
 
  1  
Reply Fri 9 Jul, 2010 09:06 am
@mickalos,
mickalos wrote:


As a philosophy undergraduate currently applying to graduate courses, you may have been correct about the dominance of platonism fifty years ago, but I assure you that this is no longer the case. The objections to platonism have mounted up, and philosophers today are more open to alternatives to the correspondence theory of truth, especially in areas like mathematics and ethics. There are a plethora of alternatives to platonism, from naturalism to quasi-realism, to suggest that platonism is correct because fictionalism is counterintuitive (though I find fictionalism no more counter intuitive than platonism) is absurd.


The diversity of views are just details of some basic ones. Math objects have three options:

1. located in the world.
2. located in the mind.
3. platonism.

2 is universally rejected. There are many views, but many of them are completely implausible, except for academic exercises.





Quote:
You make it sound as if I object to platonism on a mere whim. I object to the idea of abstract objects because the notion of a non-causal, non-contingent, non-spacio-temporal object does not make sense. In what sense can an object be said to exist, and yet have no mass, no point of creation, and no causal connection with the world or anything in it?


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



Quote:
You are also quite right to say that there is an epistemic problem as well. How can the platonist justify his knowledge that 2+2=4 when he cannot explain his relation to the objects that he posits? Indeed, as Ughaibu points out, it does not seem as if the platonist can even individuate his immaterial objects (a point also made by Strawson against substance dualism).


Is this really a problem? As a platonist, i just need to say '3 is prime' corresponds to some AO. I don ` t need to explicate the relationship between AO in plato ` s haven.

The epistemic problem of how we come to know AO is difficult. Perhaps, it is mathematical intuition advocated by Godel.



ughaibu
 
  1  
Reply Fri 9 Jul, 2010 09:38 am
@TuringEquivalent,
TuringEquivalent wrote:
I am not sure formalism captures mathematical truths. Godel show that there will be always arithmetic facts that are not deducible within the system.
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).
TuringEquivalent wrote:
there is no difficulty for different branches of math, since different areas of mathematics simply describes different classes of abstract objects in the platonic world view.
The problem for realists is the implication that some abstract objects both exist and dont exist.
TuringEquivalent wrote:
i believe in laws of nature.
I know, but your beliefs dont define my reality.
TuringEquivalent wrote:
i don` t see any problem with why a mathematical proposition cannot map into many abstract objects.
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.
Alternatively, if they claim that the only abstract objects which exist are those which correspond to true propositions, then they have two immediate problems: 1) truth becomes equivalent to existence, so they need to abandon the idea that existence isn't a property or abandon their motivation for realism about abstract objects, 2) under this view, false statements dont express propositions, so they will need to abandon reductio ad absurdum. This too will commit them to some form of constructivism.
In short, either answer to my question appears to commit the Platonist to constructivism, and the corollary of this, as far as I can see, is that the Platonist is committed to a fictionalist account of classical maths.
But the problems dont stop there. Even within constructive maths there are different formalisations, and across these contradictory theorems can be proved. How does the Platonist suggest that we find, from amongst these, which one corresponds to reality, when reality has no existence in time or space and is causally inert?
mickalos
 
  2  
Reply Fri 9 Jul, 2010 10:59 am
@TuringEquivalent,
TuringEquivalent wrote:
The diversity of views are just details of some basic ones. Math objects have three options:

1. located in the world.
2. located in the mind.
3. platonism.

2 is universally rejected. There are many views, but many of them are completely implausible, except for academic exercises.

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.

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. 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..




Quote:
You make it sound as if I object to platonism on a mere whim. I object to the idea of abstract objects because the notion of a non-causal, non-contingent, non-spacio-temporal object does not make sense. In what sense can an object be said to exist, and yet have no mass, no point of creation, and no causal connection with the world or anything in it?


I said you were bias toward AO, and favor concrete objects. You have to deal with your own personal demons. [/quote]
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. You yourself cannot give the term "abstract object" content, just try and imagine a conversation that tried:
Q. Do they exist?
A. Yes
Q. What do they look like?
A. They can't be seen.
Q. So they are invisible?
A. Kind of... you can really touch them either.
Q. Are you sure you aren't just making this up? Where do they exist?
A. In the realm of abstract objects.
Q. Where is that?
A. I don't know.
Q. Have you been there?
A. It's not exactly the sort of place one can 'be', it doesn't support spacio-temporal properties.
Q. In what sense can anything be said to 'be' other than that it was at a certain place at a certain time?
A. ...
Q. You seem to be puzzled. How about this, what would be different if they did not exist, or if they existed in a different configuration? Although, I'm not quite sure what a configuration that doesn't involve space and time is supposed to be.
A. They necessarily exist, and could not possibly exist in any other... configuration.
Q. Surely, if something can be instantiated it can also be uninstantiated?
A. That's normal objects, these are abstract objects.
Q. Aren't you begging the question?
A. Sounds like you need to exercise some demons
Q. Clearly.

Quote:

Is this really a problem? As a platonist, i just need to say '3 is prime' corresponds to some AO. I don ` t need to explicate the relationship between AO in plato ` s haven.

The epistemic problem of how we come to know AO is difficult. Perhaps, it is mathematical intuition advocated by Godel.


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.

Quote:

I am not familiar with the technical details of Emil claim, but i don` t see any problem with why a mathematical proposition cannot map into many abstract objects. The details are not relevant. In the book "platonism, and anti-platonism", the author say there need not be "uniqueness" to the mapping at all. It is consistent to maintain platonism, even tho there is not unique correspondence between math statements, and abstract objects.

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.

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.
 

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