Death Monkey wrote:

I said that it is nonsensical to say that specifying what something is does not constitute a definition

The definition of a set can, and indeed must, specify which elements are instantiated in that set. To say that this is not a definition is nonsensical, because it is nothing more than specifying what the set is.

How can I use a hammer to build a chair when hammers don't build anything?

I think we are arguing over semantics here. The point is that numbers are conceptual tools that we use when describing things.

Likewise, when you say "5 + 7 = 12", unless you specify some units or specific context, I would interpret this to be saying "according to the axioms of arithmetic, 5 + 7 = 12"

Now, maybe you didn't inted the statement that way. But then you have some other context in mind. If so, you need to specify that context. But if that context is such that the statement is one about reality, then it is not what I would consider to be a (purely) mathematical statement anymore.
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According to your own above statement, it does not make the same predictions as GR. It is a different theory.

your justification for this would have to be based on sensory information, because that is the only source of information about the world that you have.

Only because you do not recognize that you are talking about completely different statements.

The statement "parallel lines never intersect according to the axioms of euclidean geometry" is an analytic statement.

The statement "parallel lines never intersect in reality" is a synthetic statement.

I already did. One possible explanation is that mass is not a conserved quantity under the conditions of the experiment. In other words, that the rules of addition don't apply under those conditions.

We know that they do, but again this knowledge is a posteriori.

Which part are you asking about?

The underlined is a reason you provided which I showed was insufficient. In post #49 the first thing you do is to admit that I am correct, that the specification of what the set S is, is not a definition, because it makes a singular judgment. Henceforth it is not a sufficient reason that something is a specification of a set, for it to be a definition.

What does "conceptual tool" mean? A concept describes a set of characteristics necessary for a thing to be classified as such e.g the characteristics of "man" and "unmarried" are necessary for being a "bachelor" If a number is a concept, it must have certain characteristics which it describes of things.

But this is not the same statement as the underlined.

A different theory can make the same predictions as another theory e.g. "The summer will be cold and rainy" makes the same predictions as "The summer will be cold". It just makes additional predictions.

Why is this the only source of information I have?

Still you make reference to the "parallell postulate" with "it" in both cases. That is very confusing, or do you mean to say the parallell postulate is a sentence that has these two different interpretation?

Actually there are cases where mass do not conserve e.g. nuclear fission, during which mass is converted into kinetic energy. Now this types of situations would not count as refutations of 5kg+7kg=12kg in my interpretation of the statement, because putting together the masses resulting in fission is not an example of the statement in the first place. One of the things my interpretation says is that the objects measured must be represented as simultaneous (one thing that Kant made clear according to the document I provided earlier.) which prevents them from altering during the process.

Sorry, I meant about the last sentence. Why am I just talking about a new sense? In any case you seem to have defined "sense" as "source of information about reality".

If this is the case then certainly our disagreement about whether "math is a priori synthetic or not" is at least semantic. According to Kant, senses are only a specific kind of way our knowledge can immediately relate to objects, namely a passive way i.e. through being affected by them. It is by itself not able to inform us of anything but merely provide us with incontrovertible data of how it is affected by objects i.e. sensations. As an example of this. Put one hand in cold water, and the other in hot water. Then put both in the same water. One will produce a cold sensation while the other a hot sensation. Neither is wrong, because they do not represent the temperature of the water, but how we are affected by it. Now given that a posteriori/a priori distinction is based on the definition of sense, it shouldn't be surprising that we are in disagreement if you use a different definition of "sense", but this may just be a disagreement in semantics.

Death Monkey wrote:

This is incorrect. As I already explained, your attempt at a counter-example fails, because your so-called "definition" does not just specify what a particular set is, but also makes a synthetic claim.

And as I have already explained, this same criticizm is not applicable to the definition of the natural numbers, because the "singular judgements" involved there are not synthetic claims.

As I already pointed out, you are making an invalid generalization from the fact that some singular judgements are synthetic, to saying that all of the must be.

keda wrote:

"my cat is an animal" is synthetic because it is a complex of judgments one of which is "this is a cat". Even if my cat cannot be a non animal, it is a synthetic judgment, because the negation is not a contradiction, since only one of the judgments have to be false to falsify the whole statement, and if there is no cat, the statement is false. The same goes for "0 is a natural number", because if 0 doesn't exist, the statement is false.

Numbers have characteristics that can be used to describe things, obviously. The natural numbers have ordering characteristics that can be used for counting physical objects, as an example. But no physical objects need exist for them to have those ordering properties. They have them by definition. The fact that we can use numbers to count objects is not a fact about numbers. It is a fact about a relationship between numbers and physical objects. That this relationship obtains is a synthetic fact, but the definition of the natural numbers in no way asserts nor requires that such a relationship exist.

It is not the same sentence. Whether or not it is the same statement depends on what that sentence is interpreted to mean. I have just explained what I interpret the sentence to mean.

As I already stated, if you interpret the sentence "5 + 7 = 12" to mean something other than that, then you need to explain what you interpret it to mean. I am fairly certain that what you interpret it to mean is not a purely mathematical statement at all, but instead a statement about a relationship between mathematics and reality

Aside from the fact that this point has no relevance to the particular case at hand, you are quibbling about semantics here. The set of predictions made by two such theories is not the same set.

They are statements about what follows from a specified (or implied) set of axioms.

Irrelevant. There is still no a priori garauntee that the statement will hold. The only reason we even expect it to hold, is because we have observed that it does

. For that matter, if we had not made such observations, it never would have occurred to us to invent the concept of "mass" in the first place. The very claim that there is such a thing as "mass", is a posteriori.

The disagreement is not just about semantics. Even if I accept your definition of what constitutes a sense, I still cannot agree that judgements about specific relationships between mathematics and reality can be made without reference to sensory information. Introspection alone cannot tell you anything about the world beyond your mind.

Death Monkey wrote:

Yes, it is. In post 49 I said that what you were giving was an example of an attempt at a definition which also also includes the assumption that a synthetic proposition is true. So it is not just a definition. In is a claim, plus a definition.

It is a definition. It defines what you mean by the symbol "S". It is just not only a definition, because it also includes assumptions about synthetic propositions.

I did not claim that specifications are definitions.

The definition of a set can, and indeed must, specify which elements are instantiated in that set. To say that this is not a definition is nonsensical, because it is nothing more than specifying what the set is.

The fact that some concepts cannot be defined without making implicit assumptions about synthetic judgements just reflects the fact that some concepts are about the real world.

Your logic teacher is absolutely correct, as long as he/she means "existence" in the sense of being part of reality. But as I have repeatedly pointed out, the statement "0 is a natural number" does not require that 0 exist in this sense, as the statement "my cat is an animal" does.

I don't understand the question. There are many relationships between various number systems and physical objects. Every time we use mathematics to model reality, we are theorizing about such relationships.

I define it mathematically, depending on which formal system I am refering to.

Of course not. It makes absolutely no sense to say that an object is a 0.

They are not real, so by your reasoning, all mathematical statements would have to be false, which is ridiculous.

The underlined sentence is ambiguous. In order for it to mean anything, additional clarification must be provided. This clarification can be explicit, as it was when I expanded in into "according to the axioms of arithmetic, 5 + 7 = 12", or it can be implied by the context.

Likewise, when you say "5 + 7 = 12", unless you specify some units or specific context, I would interpret this to be saying "according to the axioms of arithmetic, 5 + 7 = 12

They are analytic, because the set of axioms is self-referential.

In other words, take the axioms of set theory. Each of those axioms is essentially a statement of the form "in set theory, it is true that...".

It is thus a completely self-contained formal system, which does not make any claims about, or even references to, anything real.

We have no a priori justification for thinking that mass should be conserved under any circumstances, much less all of them.

For that matter, if we had not made such observations, it never would have occurred to us to invent the concept of "mass" in the first place. The very claim that there is such a thing as "mass", is a posteriori.

You can't make any a priori synthetic judgements about it, because your only source of information about how mass works in the real world is your observations.

Nope. That is not an example of me telling you anything about the world outside of my mind. It is an example of me telling you about what I can know about the world outside of my mind. Knowing is a mental process. So in fact, I am just telling you something about my mind.

Exactly. The only access the mind has to the rest of the world is through conventional sensory input (meaning excluding your sense of awareness of your own thoughts as a "sense").

jtoma wrote:


Tidying:
1) Mathematical statements are real.
2) Mathematical statements refer to reality.
3) A statement defined to describe a physical process is not mathematical.
4) Counting can be represented by mathematical statements. (assumed)
5) The elements in a mathematical statement refer to sets or their members.
6) Sets or their members refer to physical masses.

(3) is in response to Death Monkey. (4-6) represent the last quoted sentence. Have I represented this aspect of your projected battle position fairly?

I mean, I think there could be empirical evidence that someone who thinks that his mind has access to the world via something other than sensory input does not have a better grasp of the world than someone who believes their only inputs are sensory