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).
The problem for realists is the implication that some abstract objects both exist and dont exist.
TuringEquivalent wrote:I know, but your beliefs dont define my reality.i believe in laws of nature.
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
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..
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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.
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.
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.
ughaibu wrote:Why/How?The problem for realists is the implication that some abstract objects both exist and dont exist.
The latter do not correspond to any AO.
TuringEquivalent wrote:Because incompatible theorems can be proved, within mathematics as a wholeughaibu wrote:Why/How?The problem for realists is the implication that some abstract objects both exist and dont exist.
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.
this is consistent with platonism, since, these two deductive system describes different classes of AOs.
To figure out if Fn is false, i just need to know the predicate F does not apply to the object, AO n .
ughaibu wrote:
TuringEquivalent wrote:Because incompatible theorems can be proved, within mathematics as a wholeughaibu wrote:Why/How?The problem for realists is the implication that some abstract objects both exist and dont exist.
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.
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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 .
TuringEquivalent wrote: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.this is consistent with platonism, since, these two deductive system describes different classes of AOs.
But, in the case of a proof by contradiction, you will just have meaningless statements.
TuringEquivalent wrote:
ughaibu wrote:
TuringEquivalent wrote:Because incompatible theorems can be proved, within mathematics as a wholeughaibu wrote:Why/How?The problem for realists is the implication that some abstract objects both exist and dont exist.
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.
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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.
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.
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".
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.
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.
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.
TuringEquivalent wrote: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.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.
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.
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.
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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.
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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.
ughaibu wrote:the concrete object, catgeneralisation
TuringEquivalent wrote:Even without a philosophical and mathematical training, you should be able to appreciate that a particular is not the general.ughaibu wrote:the concrete object, catgeneralisation
Is this why you think concrete objects are AO ?
TuringEquivalent wrote:What on Earth are you talking about?Is this why you think concrete objects are AO ?
if abstract objects are required for mathematical statements to be true,then they're required for any generalisation to be true.
"Any generalization" would include statements like 'cat is in the mat'.
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.
"Any generalization" would include statements like 'cat is in the mat'.
TuringEquivalent wrote:Of course it wouldn't!"Any generalization" would include statements like 'cat is in the mat'.
