keda wrote:
Now if Kant says Euclidian geometry is synthetic, which he does, then that means he thinks its negation, non eucliean geometry is at least logically possible.

Then look at it this way. The definition of a bachelor is an unmarried man. Does this mean that any unmarried men exist? No, because definitions do not define things into existence. Otherwise it is no longer a definition then, because it assumes something exists, which is a synthetic claim. It goes beyond the concept of merely being a bachelor in saying that a bachelor exists. The same is true about natural numbers. If you say something about a specific number, such as 0, it is no longer a definition, but a synthetic claim.

It is a complex of judgments, one of which is singular, namely that 0 is a natural number, as I already pointed out.

I already provided a simplified a priori synthetic arithmetic with synthetic a priori addition. Its truth relies on the ability to intuit a sequence of representations in time, in order to count them as defined by the successor operation as I defined it, or summation in gerneral. Without this ability, it cannot be said to be true.

Specifically, it uses the existential quantifier to explicate that both of the operations need third party object(s) to exist to obtain its truth value. The truth of this claim is ovely evident in that we use arithmetics purely in representation of time, such as after 5 and further 7 hours 12 in total has passed.

Well I was talking about it in that sense, even if you didn't. The article you referenced didn't suggest either way.

Euclidean geometry is just a bunch of axioms. not something to be defined.

Euclid himself believed they were statements about reality, and not empirical but self evident, so he certainly believed it was a pure science.

Synthetic means true in virtue of something further than the meaning of its constituent concepts. which means that the concepts are insufficient to determine the truth value, however a contradiction arising from them would mean it is sufficient to determine the truth value, namely that it is false. Thus its negation must be true. The negation of a contradiction is a tautology or analytic judgment, which means its negation is a contradiction. Now if Kant says Euclidian geometry is synthetic, which he does, then that means he thinks its negation, non eucliean geometry is at least logically possible.

Death Monkey wrote:

You are mixing up two completely different meanings of the word "exists". In the case of unmarried men, saying that an unmarried man exists is to say that such a man is instantiated in reality.

It is not possible for 0 to not be a natural number without redefining what the natural numbers are.

The ability to count, or even to intuit the concept of counting, depends on sensory experience.

You are talking about physics. You are talking about using mathematics to make inferences about real objects.

you cannot possibly have justification for claiming that natural numbers are instantiated in this world

You just did define it. You defined it to be a bunch of axioms. That is what it is.

They are not logically possible within the context of euclidean geometry.

Mathematical statements are analytic and a priori up until the point that you make the claim that these statements tell us something about reality. At that moment, they because synthetic. You seem to regard this step as given, as though we could not consider the statements independantly of any claims about reality, but I don't understand why you would hold this view

Nor do I understand how you could regard any claim about the applicability of a particular mathematical statement to reality as being a priori. I'm sorry, but the notion of "self evident" or "intuitively obvious" being sufficient justification, was discredited long ago. It just doesn't hold water.

There is no judgement about the applicability of any mathematical statement to reality that can be justified by anything other than empirical evidence. Anybody who claims otherwise needs to demonstrate that the justification they are citing is actually reliable and sufficient to make the judgement.

I was using existence in the more general sense of instatiation at all. 0 is an instance of natural numbers, is a synthetic judgment and you are a bachelor is a synthetic judgment, because both are singular judgments.

My point is that definitions cannot instantiate numbers like 0, because then it is no longer a definition but a synthetic claim.

How so?

Saying what something is, is not to define it. 'P is a statement' is not a definition.

Lets say P="This is a blue flower". Does that mean that P is analytic? Of course not. What you are committing is a category error. Statements and axioms are not entities to be defined like "bachelor" and "natural number" that can be defined.

I'm not at all sure what this is suppose to mean.

I'm not saying Euclides was right in his view, even though one could say he believer they were true synthetic a priori, but I'm not using his views as justification, just pointing out that he did not define the axioms in an analytic way.

It is more reliable and justifiable than any empirical evidence, because it is something that must be presupposed for there to be any empirical evidence in the first place.

The set S contains only my cat, specifies what the set is, but it is not a definition, because it makes a singular judgment about there being a cat.

Thats a very complex way of saying that numbers are not real, but I don't see any arguments for this.

This got to be the most desperate argument you've come up with so far. When a math teacher demonstrates a proof on the whiteboard, it an a posteriori truth because you have to look at the whiteboard?

If anything this would be self defeating argument because math would be a posteriori and therefore not analytic. Surely you didn't mean to say that reasoning is a posteriori?

That is just naming the set of axioms.

In any case, if you are to define definition in such a broad way, you cannot say definitions are necessarily analytic truths because defining "Molly" to refer to my cat, presupposes the existence of a cat.

Granted although as such I don't see the relevance.

If so I would type "A1 /\ A2... An => 5 + 7 = 12", and not just "5 + 7 = 12",

and I don't agree with your definition of mathematical statement.

You need mathematics because all objects of experience must be located in space and time. Even if this is done subconsciously without you thinking about it, it is necessary underlying framework for justifying the objective validity of your experience, and when for some reason this subconscious process is tricked and creates an illusion, you need to do the dirty work yourself, such as when a pen put in a glass of water appear not straight, you must apply optical physics along with geometry to conclude it in fact is straight.