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Wed 16 Jun, 2010 10:40 pm
From the posters on these boards, I've seen three arguments in support of realism about abstract objects.
1) mathematics is essential in scientific theories, we should be committed to the reality of all objects essential to scientific theories, therefore we should be committed to the reality of mathematical objects.
The main objections to this are:
a) mathematics has not been demonstrated to be essential in scientific theories
b) science doesn't purport to generate truth, it purports to be adequate, therefore we are only committed to an adequacy of mathematics.
2) for mathematical statements to be true, they must be correct descriptions of an extant state of affairs, therefore there is a state of affairs corresponding to any mathematically correct statement.
Again, for the moment, just two objections:
c) the truth of mathematical statements is exactly that which has been defined as true, so, mathematics is self contained and its statements needn't be true of anything but themselves
d) all animals respire, therefore "an animal respires" is a true statement. This gives the following argument:
i) an animal respires
ii) for i to be true, there is a non-specific abstract object that is an animal which respires
iii) abstract objects are causally inert
iv) respiration is causally active
v) therefore, either no animal respires or there is a non-specific truth without the requirement for an abstract object.
3) if abstract objects exist, then we can solve various explanatory problems, we want to solve these problems, therefore we should hold that abstract objects exist.
Again two objections:
e) supernatural solutions are not satisfactory and realism about them shouldn't be adopted for its psychological advantages
f) unless we can demonstrate that explanatory completeness is available to humans, we have no reason to suppose all problems have solutions, and as explanatory completeness appears to be impossible for humans, we should not adopt beliefs that assume the truth of explanatory completeness.
If you think that realism about abstract objects is supportable, give it a go.
@ughaibu,
Quote:a) mathematics has not been demonstrated to be essential in scientific theories
I don't see how the scientific revolution, comprising Galileo's cosmology, Cartesian algebraic geometry, and Newton's laws of motion could have occurred without mathematics, let alone Einstein's theory of relativity.
Perhaps I should turn the question around, and ask you how Newton, Galileo and Einstein would have proceeded without mathematics. Any suggestions?
Quote:the truth of mathematical statements is exactly that which has been defined as true, so, mathematics is self contained and its statements needn't be true of anything but themselves
The fact is, though, that in the history of science, many significant discoveries have been made solely due to mathematical reasoning which has predicted experimental results that were subsequently verified by experimental observation. This is especially true in the realm of subatomic physics. So if all such statements were literally tautological, why did they need to be verified by observation, and why did we not know already of the results which they predicted?
@ughaibu,
ughaibu wrote:
From the posters on these boards, I've seen three arguments in support of realism about abstract objects.
1) mathematics is essential in scientific theories, we should be committed to the reality of all objects essential to scientific theories, therefore we should be committed to the reality of mathematical objects.
The main objections to this are:
a) mathematics has not been demonstrated to be essential in scientific theories
b) science doesn't purport to generate truth, it purports to be adequate, therefore we are only committed to an adequacy of mathematics.
2) for mathematical statements to be true, they must be correct descriptions of an extant state of affairs, therefore there is a state of affairs corresponding to any mathematically correct statement.
Again, for the moment, just two objections:
c) the truth of mathematical statements is exactly that which has been defined as true, so, mathematics is self contained and its statements needn't be true of anything but themselves
d) all animals respire, therefore "an animal respires" is a true statement. This gives the following argument:
i) an animal respires
ii) for i to be true, there is a non-specific abstract object that is an animal which respires
iii) abstract objects are causally inert
iv) respiration is causally active
v) therefore, either no animal respires or there is a non-specific truth without the requirement for an abstract object.
3) if abstract objects exist, then we can solve various explanatory problems, we want to solve these problems, therefore we should hold that abstract objects exist.
Again two objections:
e) supernatural solutions are not satisfactory and realism about them shouldn't be adopted for its psychological advantages
f) unless we can demonstrate that explanatory completeness is available to humans, we have no reason to suppose all problems have solutions, and as explanatory completeness appears to be impossible for humans, we should not adopt beliefs that assume the truth of explanatory completeness.
If you think that realism about abstract objects is supportable, give it a go.
For mathematical statements to be true, it has to be about something. What is these something? It is abstract objects.
abstract objects is essential to physics. In physics, if the fundamental equation processes certain symmetries, it corresponds to a set of conservation laws. What are these symmetries? These symmetries are about physical systems that could be described by abstract objects. Thus, abstract objects exist.
@jeeprs,
jeeprs wrote:Perhaps I should turn the question around, and ask you how Newton, Galileo and Einstein would have proceeded without mathematics. Any suggestions?
I again refer you to Field and Balaguer.
jeeprs wrote:So if all such statements were literally tautological, why did they need to be verified by observation, and why did we not know already of the results which they predicted?
There is nothing here that supports realism, any more than the fact that I can look at a map and make a prediction about how long it will take me to walk somewhere. How does this imply the existence of abstract objects? You need to offer an argument, not ask me questions with the apparent assumption that abstract objects exist by default, they dont. Realists tell me that unicorns dont exist, and by this they mean that there are no unicorns with any location in space or time, but that is exactly what is said about abstract objects, they have no location and are causally inert. This means that they have to be argued for, not assumed to have default status, because they have the exact same description as the nonexistent.
@TuringEquivalent,
TuringEquivalent wrote:For mathematical statements to be true, it has to be about something. What is these something? It is abstract objects.
I have no reason to believe that, as I illustrated with the respiring animal, I have good reason to reject it.
TuringEquivalent wrote:abstract objects is essential to physics. In physics, if the fundamental equation processes certain symmetries, it corresponds to a set of conservation laws. What are these symmetries? These symmetries are about physical systems that could be described by abstract objects. Thus, abstract objects exist.
You'll need to structure this argument more clearly. I dont see how you've established that abstract objects are essential to physics, (if this is true, presumably all physicists are realists about abstract objects) or that they exist.
Oh, good grief.
Well, I'll try to read this.
Hasta manana.
@ughaibu,
ughaibu wrote:
TuringEquivalent wrote:For mathematical statements to be true, it has to be about something. What is these something? It is abstract objects.
I have no reason to believe that, as I illustrated with the respiring animal, I have good reason to reject it.
TuringEquivalent wrote:abstract objects is essential to physics. In physics, if the fundamental equation processes certain symmetries, it corresponds to a set of conservation laws. What are these symmetries? These symmetries are about physical systems that could be described by abstract objects. Thus, abstract objects exist.
You'll need to structure this argument more clearly. I dont see how you've established that abstract objects are essential to physics, (if this is true, presumably all physicists are realists about abstract objects) or that they exist.
Let me ask you a question! Do you think "3 is prime" is true? If so, what makes it true?
Physicists are all realist about abstract objects. If they don` t, it is because they don` t know how to say what they believe. To borrow an example from geometry. The triangle has the symmetry operation of 0, 120, 240, 360.
Why do we suppose there are these symmetry operations unless there is object called 'triangle'? We presuppose there are these objects when we make statements about them.
@TuringEquivalent,
TuringEquivalent wrote:Do you think "3 is prime" is true? If so, what makes it true?
I dont think anything "makes" it true, it's true by virtue of the definitions and known to be true by observation. How would an abstract object "make it true"?
TuringEquivalent wrote:Physicists are all realist about abstract objects.
I see no reason to accept that, and even if true, you haven't given an argument, just stated the generality of a belief.
@ughaibu,
ughaibu wrote:
TuringEquivalent wrote:Do you think "3 is prime" is true? If so, what makes it true?
I dont think anything "makes" it true, it's true by virtue of the definitions and known to be true by observation. How would an abstract object "make it true"?
TuringEquivalent wrote:Physicists are all realist about abstract objects.
I see no reason to accept that, and even if true, you haven't given an argument, just stated the generality of a belief.
How is "3 is prime" true by definition? Did some one go to all the trouble of defining all the primes in the number line? Suppose it is true by observation. What are we observing to see that "3 is prime" is true?
What i wanted to show is that mathematicians, and physicists make claims. Those claims presuppose the existence of abstract objects.
@TuringEquivalent,
TuringEquivalent wrote:How is "3 is prime" true by definition?
What are we observing to see that "3 is prime" is true?
You know the definition of "three" and you know the definition of "prime". Observe:
111
(1)(1)(1)
(11)(1)
(111)
@ughaibu,
I don't know who Field and Balaguer are. Perhaps, if you are not too busy, you might consider answering the question yourself.
Furthermore, my response to your questions did not posit abstract objects or use the term 'realism'. I said "the fact is, though, that in the history of science, many significant discoveries have been made solely due to mathematical reasoning which has predicted experimental results that were subsequently verified by experimental observation. " Are you able to refute this? Do you think it is wrong to infer from this that mathematics will produce results that are more than tautological?
@ughaibu,
ughaibu wrote:
TuringEquivalent wrote:How is "3 is prime" true by definition?
What are we observing to see that "3 is prime" is true?
You know the definition of "three" and you know the definition of "prime". Observe:
111
(1)(1)(1)
(11)(1)
(111)
The smellest prime is 3, and not 1.
How is what you say, true? Why is 111, 3? Why is it not ()()()? You can ` t say it is simply marks on paper, since, it would be meaningless. You got to have a model for whatever formal system.
@jeeprs,
jeeprs wrote:
I don't know who Field and Balaguer are. Perhaps, if you are not too busy, you might consider answering the question yourself.
Furthermore, my response to your questions did not posit abstract objects or use the term 'realism'. I said "the fact is, though, that in the history of science, many significant discoveries have been made solely due to mathematical reasoning which has predicted experimental results that were subsequently verified by experimental observation. " Are you able to refute this? Do you think it is wrong to infer from this that mathematics will produce results that are more than tautological?
Mathematics don ` t make claims about the world at all. What you mean is physics. Physicists uses mathematics to describe, model the world. You shouldn` t confuse the two...
@TuringEquivalent,
TuringEquivalent wrote:Why is 111, 3?
Does this mean that you dont understand the definition of "three"?
Your line of questioning seems to me to be aimless. You haven't dealt with the counter example of the respiring animal, you haven't argued for the claim that abstract objects are essential for physics and you haven't offered an explanation of how abstract objects "make true". Have you given up on these?
@jeeprs,
jeeprs wrote:I don't know who Field and Balaguer are.
Well, you certainly should, as it's only a week or so since I linked to Balaguer's publication page on North's atheist thread.
jeeprs wrote:Are you able to refute this? Do you think it is wrong to infer from this that mathematics will produce results that are more than tautological?
As far as I recall, the only way that you've shown mathematics to be useful is for measurements. As far as science goes, those measurements must refer to physical observables. In other words, what you are saying is that we can measure physical objects, and as physical objects are not abstract objects, you have not offered any argument that implies the existence of abstract objects. Neither have you demonstrated non-tautological mathematics, what do you think such a thing would be?
@ughaibu,
ughaibu wrote:
TuringEquivalent wrote:Why is 111, 3?
Does this mean that you dont understand the definition of "three"?
Your line of questioning seems to me to be aimless. You haven't dealt with the counter example of the respiring animal, you haven't argued for the claim that abstract objects are essential for physics and you haven't offered an explanation of how abstract objects "make true". Have you given up on these?
You are presupposing that 3 has a definition! Does dog, or cat have a definition? What about the sun? The word 'sun' had a referent, and that referent is sun. Similar, i think it does not make sense to say '3' had a definition. It is a categorical error. The names of things don` t have definitions. They have referents.
What i gave you is the following.. When physicists make claims about the world. Those statements presuppose the existence of abstract objects to be true. I gave you two examples. The first example to about symmetry. Is that an argument? Yes. Perhaps you don` t know what 'argument from best explanation' is. Do you want a link from wikipedia?
@TuringEquivalent,
TuringEquivalent wrote:When physicists make claims about the world. Those statements presuppose the existence of abstract objects to be true. I gave you two examples.
Your examples dont require the presupposition, they just illustrate that such a presupposition is possible. How about the fact that mathematical models which exactly match the phenomenal world are impossible, even in principle? Which do you think the physicists believe, the abstract objects are the reality or the phenomenal world is? How about the problem that different consistent mathematical systems and logics can each prove theorems which would be contradictory in a different system or logic? Do you think that there is some real, possibly undiscovered, mathematical system that describes the world? That the real logic is classical, independence friendly, relevance, universal or something else? How would you decide which is the correct logic, etc? Defending abstract objects isn't a matter of asking a few questions about the number three, it is a seriously difficult story.
@ughaibu,
ughaibu wrote:
TuringEquivalent wrote:When physicists make claims about the world. Those statements presuppose the existence of abstract objects to be true. I gave you two examples.
Your examples dont require the presupposition, they just illustrate that such a presupposition is possible. How about the fact that mathematical models which exactly match the phenomenal world are impossible, even in principle? Which do you think the physicists believe, the abstract objects are the reality or the phenomenal world is? How about the problem that different consistent mathematical systems and logics can each prove theorems which would be contradictory in a different system or logic? Do you think that there is some real, possibly undiscovered, mathematical system that describes the world? That the real logic is classical, independence friendly, relevance, universal or something else? How would you decide which is the correct logic, etc? Defending abstract objects isn't a matter of asking a few questions about the number three, it is a seriously difficult story.
When a physicists say "LOOK, there is a circular table in that direction!". He does not mean the table is a circle in the mathematical sense. We are strictly talking about physical models of the real world, and they are mathematical. When newton postulated F= G mM/r^2, it describes a structure. There is set of properties associated with that structure. This is surely mathematical!
Your musing about logic makes no sense to me at all. The traditional debate between platonism and nominalist don` t at all concern itself with logic. Logic is a tool. There is no ontological meaning to logic. It is a normative discipline.
@TuringEquivalent,
TuringEquivalent wrote:He does not mean the table is a circle in the mathematical sense. . . . . When newton postulated F= G mM/r^2, it describes a structure. There is set of properties associated with that structure. This is surely mathematical!
You still appear to be talking about beliefs, and not giving any argument. Let's assume that Newton believed that there was some abstract object which made his statement true, in this case, why is his statement superseded by Einstein? Newton's statement was superseded as a statement about the phenomenal world, not as a statement of mathematics. If it were only a statement about mathematics, and it was true due to corresponding to some abstract object, this would be of no interest to physicists, because physicists are scientists attempting to generate useful statements about the phenomenal world.
TuringEquivalent wrote:There is no ontological meaning to logic. It is a normative discipline.
The realist argument includes all manner of abstract objects, and your above comment could equally be applied to maths as to logics.
@ughaibu,
ughaibu wrote:
TuringEquivalent wrote:He does not mean the table is a circle in the mathematical sense. . . . . When newton postulated F= G mM/r^2, it describes a structure. There is set of properties associated with that structure. This is surely mathematical!
You still appear to be talking about beliefs, and not giving any argument. Let's assume that Newton believed that there was some abstract object which made his statement true, in this case, why is his statement superseded by Einstein? Newton's statement was superseded as a statement about the phenomenal world, not as a statement of mathematics. If it were only a statement about mathematics, and it was true due to corresponding to some abstract object, this would be of no interest to physicists, because physicists are scientists attempting to generate useful statements about the phenomenal world.
TuringEquivalent wrote:There is no ontological meaning to logic. It is a normative discipline.
The realist argument includes all manner of abstract objects, and your above comment could equally be applied to maths as to logics.
I said F=GmM/r^2 is a structure with a set of properties. Do you need an argument for that? One property is translational symmetry by the transformation r--> r+a. Do you need a proof for this? I can give it to you.
Why was newton law superceded? Well, because Einsteins mathematical model is more true to reality than newton ` s F=GmM/r^2. The abstract object that Einstein postulated is more true to reality than Newton` s abstract object. Any physical model in physics are mathematical in nature. It describes an abstract object. When the physical theory is good, the abstract object in question is a good model for the real world.
Again, logic is a normative science. Traditionally, Platonism do not include logic.
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