Best argument for Platonism.

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.

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 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.

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.

Your whole point is that mathematical truth is grounded on mathematical practices.

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.

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.

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.

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

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.

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?

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?

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.

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?

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.

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

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 x2, 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?

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.

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).

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

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.

i believe in laws of nature.

i don` t see any problem with why a mathematical proposition cannot map into many abstract objects.

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.

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?

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.

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.