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It is a huge topic, but it is time to begin blogging my way into it as part of my protracted campaign against physicalism/materialism/naturalism in all its forms and wherever it may hide.
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Related Posts (on one page):
As you point out, the number of possible sets is infinite. And as you say, the members of sets can themselves be sets.
What, then, about the set of all sets that do not contain themselves as members? What is the ontological status of that teratological entity?
How does that thought now qualify as an "object"? Who's in charge of that?
Finally, did you know that the word "set" is the word that has the longest entry in most dictionaries?
What a about a world that was a true monad. Indivisible. Would these objects exist?
Also, it seems to me that these objects are mental in the sense that they are a result of they way we perceive the world. The only reason we conceive of numbers-degree of difference or enumeration- is because we perceive the world as a multiplicity and need some way to account for the realtionships between things. Admittedly once differences are postulated every possible world with differences has to include some self-consistent system of enumeration that will follow the same universal set of given rules, but it doesn't seem to me like this system is either necessary or outside our own mental processes.
The point of my comment was to defend the view that they are actually only thoughts that we have.
That problem was solved long ago. For that you must look into axiomatic set theory. In short, under appropriate axioms you cannot define such a set.
In any case, I don't think that's really germane to the post.
Who are we kidding?
Anyway, since the conceptual world is static we cannot simultaneously conceive of the properties of objects and the relationship between those properties. Maybe you have noticed that mathematics is totally propertyless, it works only within an assumed homogenous substance. It only deals in differences, not properties. The efficacy of physics comes from the fact that although we conceive of the relationships and the properties separately, we can combine them when we take them back out of the mental world apply them to what we have perceived in the physical world.
To come up with some physicalistic account of this it seems to me we have to go back to neuroscience.
I think the reason that properties can be changed around so easily is that they are stored in a separate part of our brain then our perceptions, as individual pieces of information that we have the ability to "play" with when combining them. This is just a supposition from observing my own way of thinking and doing some research on learning. I have never read anything on how mathematicial information might be stored but if it is stored in a completely separate part of the brain from the properties this might account for the reason why these are so different.
What really gets me with all this is how our brain can be set up to figure out all the rules by which mathematics works. The rules themselves are completley determinate. While we can choose our symbols we cannot choose how to relate the symbols. It's like in one part of the brain the realtionships are chooseable and in the other the objects are chooseable.
In mathmatics what we seem to have is an ability to compare. So in mathematics we seem to have infinite ways to express relationships.
I didn't think you were claiming that mathematical objects are perceptual objects, at least not in the "same way" (that Wittgensteinian problematic, again) that physical objects are perceptual, or perhaps constructed by idealization from perceptual objects (this thing called 'idealization' is another hornets' nest).
However, I don't think that perception is always and only of changing properties -- else there would be no problem of perceptual constancy. And logic/mathematics is not all nouns -- we need operators as well as operands.
The problem I was trying to highlight is that however nicely we can effect mappings between the physical world and mathematics, that such mapping is possible doesn't help us to understand how it is that mathematical objects seem to possess their properties in ways very different from the way physical objects possess their properties. It should be one of the first orders of business for those who claim mathematical objects are physical to explain (or explain away) the seeming difference.
Finally, mathematicians mostly insist they "discover" new maths, they don't invent them. They are mind-independent and (more arguably to me) physical object-independent.
What mathematicians insist about their discoveries isn't necessarily the final word on the subject, I'd say. They might simply be wrong. Also, you say they "mostly" insist; what then is the source of their disagreement, and how do we know who is right?
Bob, I agree that mathematical objects don't possess their properties in the same way physical objects do. What I was suggesting is the possibility that mathematical objects are thoughts that we have. I am certainly not claiming that mathematical objects are physical objects, any more than the rest of our thoughts are. The physicalist interpretation, I suppose, would be that our thoughts are states of our neuroanatomy.
Posted too soon. (There's probably some sort of ointment that prevents that.)
To refute someone who claims that 4 is prime, I'd ask him simply to explain his thinking on the matter - what he means by "4", by "prime" and what his idea of mathematics is generally.
Keep in mind that non-Euclidean geometry, the calculus of infinitesimals, and transfinite and imaginary numbers used to seem as plainly wrong as "4 is prime."
On the second point, I was thinking that the ability to convince everyone that there is a correct view of a mathematical proposition points to its objective status. As for non-Euclidean geometry, calculus et.al: were they thought of as wrong or just not discovered yet? Was there a long-lasting community of dissenters to these ideas?
The problem with using your being able to convince everybody of the correctness of a mathematical proposition as proof of its ontological mind-independence is that everyone around you (in these parts, at least) has, basically, the same kind of mind. In the same way, I would have no trouble establishing here in New York City the ontological certainty of the proposition "George Bush is an idiot."
Celinda, in a world that contained only one object, there would still be an infinite number of sets. In fact, in a world of no objects there are an infinite number of sets. Obviously there is the empty set, {}, there is also the set ONE containing the empty set, {{}}. There is also the set TWO containing the empty set and ONE, {{},{{}}}, etc.
Malcolm, George Berkeley's criticisms of calculus were specifically about Newton's calculus, and Berkeley's criticisms were valid. Newton's formulation of the calculus was sloppy and did not hold up to inspection. No amount of "proving itself" would ever have made Newton's calculus acceptable to rigorous mathematicians.
Eventually Newton's calculus was thrown out and replaced by the theory of limits, which was not vulnerable to Berkeley's criticisms.
That's right. We all agree that statements like '7 is prime' are true. What makes them true? What must the world be like for them to be true? Must there be something outside the mind and language for them to be about? If yes, could those things be physical? The point of my little argument was that, if there must be something extramental to serve as truth-makers for math truths, then those truth-makers cannot be physical: math objects far, far outrun physical objects.
If there is even one set, then there is an infinity of sets.
Good response to Celinda. But if 'object' is used to cover absolutely everything, then it covers sets as well. It follows that in a world with no objects, there would be no sets, not even the empty or null set. One might take this to show that there cannot be a possible world in which there is nothing at all.
Perhaps you are using 'object' to cover non-sets.
Your suggestion seems to be that there is no mathematical reality. But I take it you think there is a physical reality. Is that what you are saying?
You are missing the point of the argument inasmuch as I was assuming that there is something external to the mind that math statements are about, and then arguing that they cannot be physical objects. You are questioning that assumption. But I dealt with that in an earlier post. In other words, there are two distinct questions:
Q1. Do math statements correspond to anything extralinguistic/extramental?
Q2. If yes to Q1, then: are those extramental things physical or Platonic or something else?
I was dealing with Q2 in the present post.
My answer to Q1. is no, because numbers, sets and all of mathematics are symbols and the manipulation thereof according to certain rule sets. Some of mathematics manipulates symbols according to rules that produce useful results--a bridge that stands for centuries. Some math has no necessary relation to the physical universe--the properties of infinities, perhaps.
Imagine a universe identical to ours but with no human beings. Where are the 'prime numbers,' the 'number 4' and 'sets?' In such a universe there are only physical objects. We created these concepts and without us they have no meaning.
Therefore numbers cannot be identified with physical objects whence it follows that not everything is a physical object!
I agree heartily with all of the statements in the post up to the very last six words, the truth of which depends on the definition of (every)'thing.' Here is the heart of the question.
In some sense the univserse is just a soup of myriad whirling particles, which are simply denser in some places than others. Our minds abstract objects and create boundries out of this stuff. These boundaries define physical 'things.' I would argue minds also create the relationships that are concepts. If a relationship is a 'thing' then not everything is a physical object. That's the crux, in my understanding.
Chess tournament today and tomorrow. But I hope to respond to (some) of these very good comments. Thanks and enjoy your weekend.
“numbers, sets and all of mathematics are symbols and the manipulation thereof according to certain rule sets.”
IIRC, Gödel’s incompleteness or undecidability theorems were designed precisely to disprove this.
"Our minds abstract objects and create boundries out of this stuff."
If this is so, wouldn’t it also be true that not only are relationships not physical things, but there are no physical things. If there are no physical things, if things are merely conventional, then that is a problem for physicalism, I would think. Especially so for fundamental physical entities that are held to exist by virtue of mathematical models such as the infinite dimensional Hilbert space where quantum mechanics lives.
Just back from a tiring college-visiting excursion...
I didn't say either of those things! That was Robert.
Too tired to comment at the moment.
If this is nothing but an arbitrary game of symbol manipulation, then why should I have any confidence that I can go to the story, buy 20 cans of soda then have two sodas for each guest?
It can't be an arbitrary game. The rules of the game must be the correct rules. They have to be the rules that will give the correct answers. The question is, how do we know what the correct rules are? I say that the correct rules are the rules that always produce true statements about numbers from true statements about numbers.
If there are no numbers, then there are no true statements about numbers.
Briefly for now: Henry, there certainly are physical things. My argument is that we essentially create the boundaries between them and then give what we consider to be entities a name, after which they have a separateness to them that's only in the eye of the beholder. Example, a 'cloud' in the sky. It may seem to have sharply defined boundaries from the ground, but it doesn't.
But upon further consideration I think that I was wrong on Bills' Q1., even if my points were good...more later; must go.
To expand a bit on one thing I said: the existence of some physical entities is inferred only through a mathematical formalism. One of these is the quantum wave function. The space in which the formalism works is not only infinite dimensional in some cases, but is complex in the sense of the ordered pair (a, ib). We can't seem to form any clear idea of what this entity is in a physical sense other than through an abstract formalism.
PS. A paper relevant to this topic is:
On the Necessary Existence of Numbers
Neil Tennant
available online
Bill,
What if the universal, unchangeable nature of mathematics is caused by the way we are made ie. the "hard wiring" in our brain? Would this constitute be an abstract object under your definition or would they be wholly mental objects?
But having spent most of my lunch time on that one I now have more food for thought than I can easily digest quickly...But in regard to the current post, I believe I have an example that might support the last sentence, the truth of which I stated hung on a definition of 'thing.'
Concepts (including mathematics) can effect physical change in the world; therefore, they are 'things.'
A good example are those drawings designed to fool the eye--one looks and sees a 'young lady,' then someone simply says the words 'old lady' and you see the picture entirely differently! A concept has affected a physical change in the brain, as far as I can tell. It wasn't the mere vibrations of air in your eardrums that caused the new perception, it was an idea.
The use of mathematical concepts like 'number' and 'set' seem to operate in a similar fashion, as non-physical objects.
However, it isn't at all obvious that this happens because the concept does (or even possibly can) cause a physical change in the brain (except, again, mediated by the mind). You should look through Bill's archives of the last couple of months for some more food for thought).
What suddenly occurs to me is that this observation pretty much prevents a mind/body physicalist from being a realist with respect to abstract objects. If he were, he would have to explain how abstractions can have direct physical influence on the brain.
Bill: has anyone ever addressed this point?
I've been "offline" for a few days and missed some interesting discussion here.
As for Robert's comment just above, the physicalist response would be that it in fact WAS the vibrations of air in the eardrums. They pushed the brain into a different state, and that new state in turn caused relevant output. "Idea" is just a high-level description of this underlying process.
Perceiving an ambiguous optical illusion one way or another merely means that the same input is capable of driving the brain into two or more states. Which one is chosen presumably depends on inner or outer context.
2. Disallowing comments from a particular person, or deleting an offensive, off-topic, or otherwise substandard comment, has nothing to do with censorship. People who think otherwise confuse censorship with lack of sponsorship. I am under an obligation not to interfere with anyone's exercise of legitimate free speech rights. But I am not under any obligation to aid and abet anyone's exercise of free speech rights, legitimate or illegitimate.
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