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[Fafner88]

I don't dispute such sentences make sense, it is a question of what would show them to be true or false, that is to say whether they are true only by definition or whether they also make a claim about external reality.

Obviously they do make claims about the external reality, don't they? You agreed that "there are cats in Australia" is not simply a definition but it's truth depends on how the world is.

The problem with the geometrical straight line is that it can't exist as any particular example. A line in geometry has length but no width. Nothing in the world of objects can only have one dimension.

I doesn't matter if there are lines with no width in reality. Of course geometrical concepts are abstract idealizations, but things in reality can very closely approximate these idealizations. If I draw a very accurate triangle on a page then obviously it will exhibit to a very high degree of approximation the properties that geometry says triangles have (the sum of the angles will be close to 180 degrees, Pythagoras's theorem will be true about it etc.). It will be very strange to say that the properties of actual triangles have nothing to do with the properties of triangles described by geometry. The opposite is true: triangles in reality have these properties because they are the necessary properties of any possible triangle, and these properties can be known a priori to hold of any triangle without examining all possible triangles.

And besides, physics also talks about abstract idealization which don't (and also can't) exist in reality (frictionless surfaces, black bodies, closed systems, ideal gasses, perfect inertia etc.), but this of course doesn't prevent physics from telling us true things about the world.

To put it another way, they are the only truth available to us poor doubting philosophers! As I outlined back in post 31, Kant would argue that since these are the only sort of truth available then this is what we should understand as 'synthetic a priori' knowledge, but that is a different understanding to the conventional (?) one. As Spectrum remarks, in our exchange there is confusion because we drift between the two.

So you do agree that geometry and mathematics are synthetic a priori? Because earlier you said they are analytic.

[Londoner]

Obviously they do make claims about the external reality, don't they? You agreed that "there are cats in Australia" is not simply a definition but it's truth depends on how the world is.

It depends on both. Assuming our definitions of 'cats' and 'Australia' included the claim they are 'real' , and their connection in reality was also fact, then the claim would be true. I'm not sure what point we are discussing now.

I doesn't matter if there are lines with no width in reality. Of course geometrical concepts are abstract idealizations, but things in reality can very closely approximate these idealizations. If I draw a very accurate triangle on a page then obviously it will exhibit to a very high degree of approximation the properties that geometry says triangles have (the sum of the angles will be close to 180 degrees, Pythagoras's theorem will be true about it etc.). It will be very strange to say that the properties of actual triangles have nothing to do with the properties of triangles described by geometry. The opposite is true: triangles in reality have these properties because they are the necessary properties of any possible triangle, and these properties can be known a priori to hold of any triangle without examining all possible triangles.

I would not agree that there are such things as actual triangles, not in the geometric sense. We can see this because what is true of geometric triangles will never be quite true of triangular shaped things. I think that something being always true in geometry, but never true in reality, indicates a significant difference!

And besides, physics also talks about abstract idealization which don't (and also can't) exist in reality (frictionless surfaces, black bodies, closed systems, ideal gasses, perfect inertia etc.), but this of course doesn't prevent physics from telling us true things about the world.

But those terms you list do not describe actual objects; abstracts are just that; abstractions. Just as in maths we can talk about the abstraction 'infinity', but 'infinity' is not the name of a particular number.

As I have mentioned before, if an abstract idealisation told us true things about the world, then the Ontological Argument for God would work. We can frame the idea of God as an abstract ideal - so must we conclude that that idea of God is a 'true thing about the world'?

Kant argued that all attempts to find a 'synthetic a priori' in the sense of a proposition that was derived from reason (rather than experience, which we cannot trust) yet that could tell us a 'true thing about the world', were all versions of the Ontological Argument - an attempt to reify our abstractions, i.e. treat our abstractions as if they were things.

So you do agree that geometry and mathematics are synthetic a priori? Because earlier you said they are analytic.

This is what I mentioned in my opening post (31) and again in my last post (45). Kant would say that their truths should be regarded as 'synthetic a priori', but this was a contradiction of the earlier understanding of the idea.

[Fafner88]

It depends on both. Assuming our definitions of 'cats' and 'Australia' included the claim they are 'real' , and their connection in reality was also fact, then the claim would be true. I'm not sure what point we are discussing now.

The point is that there needn't to be any particular dog to make "dogs bark" true and by analogy there needn't be any particular line for "the shortest distance between two points is a straight line" to be true. This is to show that your original argument was unsound:

By contrast, propositions about reality are propositions that must involve measurement and location. London is at these co-ordinates; it is at this distance from New York. Such propostions can be true or false, unlike your example.

I would not agree that there are such things as actual triangles, not in the geometric sense. We can see this because what is true of geometric triangles will never be quite true of triangular shaped things. I think that something being always true in geometry, but never true in reality, indicates a significant difference!

I see no reason to accept this. Isn't a round plate shaped like a circle? Of course it's not a perfect circle but it is still has the property of being circular. On your view it must be that when we say that a circle in geometry is circular and when we say that a plate is circular we are assigning two completely different properties, but this seems to me absurd. Plates are circular precisely because they approximate circles in geometry. something doesn't have to be a perfect triangle in order to be considered a triangle, at least I don't see any argument for thinking otherwise.

But those terms you list do not describe actual objects; abstracts are just that; abstractions. Just as in maths we can talk about the abstraction 'infinity', but 'infinity' is not the name of a particular number.

Now you are simply contradicting elemental physics. Why something can't be approximately like an ideal gas for example? "approximately true" or "or approximately alike" are a perfectly sensible and meaningful ideas. If scientific idealizations don't describe reality, then why scientists are bothering with them?

As I have mentioned before, if an abstract idealisation told us true things about the world, then the Ontological Argument for God would work. We can frame the idea of God as an abstract ideal - so must we conclude that that idea of God is a 'true thing about the world'?

No, it doesn't follow that we can prove God. The abstract concept of a triangle doesn't say that there are in fact triangles, it only says that if something is a triangle then it has such and such properties, so the comparison with the ontological argument seems to me irrelevant.

This is what I mentioned in my opening post (31) and again in my last post (45). Kant would say that their truths should be regarded as 'synthetic a priori', but this was a contradiction of the earlier understanding of the idea.

I wasn't asking what Kant said, I was asking whether you agree with me or not that mathematics and geometry express synthetic a priori truths about the world, and not simply being true by linguistic convention. And if not then why?

[Wayne92587]

Woody wrote; To illustrate what I mean, take the following synthetic a priori argument:

1. Dragons have fire in their belly. 2. Fire is hot. Therefore, dragons have something hot in their belly.

This argument is logically so similar to the above bachelor argument that it might as well be the same. There is, however, a significant difference. We know from experience that bachelors exist; so, when we draw conclusions about bachelors, we can refer to certain basic facts about real things. We do not, however, know that dragons exist, and many would argue that dragons _don't_ exist. Accepting that dragons are imaginary, what this second argument tells us is about the nature of dragons. Yet, if the dragon is imaginary, then the fire in the dragon's belly is also imaginary. The second premiss deals with what we know about real fire; except that there is nothing to say that imaginary fire must be like real fire. The imaginary fire in a dragon's belly might very well be cold! Because it doesn't exist, there are no restrictions to the properties we can assign to it.

Based on my limited discussion here, I would conclude that a priori arguments by themselves do not convey any real knowledge. Real knowledge must be buttressed by empirical observation.

Comments?

Woody

Dragons are Real, existing as a beast like creature that spews Fire; the source of the Fire being the belly of the Beast, the Name of the Beast, Dragon, the person who's Passion has been perverted, distorted, having to much Fire in his or her Belly is a Man Beast, is likened to an angry sergeant spewing his charges at us.

[Fafner88]

This argument is logically so similar to the above bachelor argument that it might as well be the same. There is, however, a significant difference. We know from experience that bachelors exist.

What 'bachelor argument' do you mean? The claim was that it's possible to know that bachelors are unmarried man simply by understanding what the terms mean without conducting any further empirical investigation, but I don't see what your dragon example has to do with this.

[Belinda]

Wayne92587 wrote:

I would conclude that a priori arguments by themselves do not convey any real knowledge. Real knowledge must be buttressed by empirical observation.

You are quite right. A priori propositions are analytic whereas inductive (empirical) propositions are synthetic .

However what I have just written depends for its truth on 'knowledge' meaning progressive knowledge. Real life decisions and the knowledge acquired from them are most often synthetic/empirical/ inductive.

[Londoner]

Faffner

The point is that there needn't to be any particular dog to make "dogs bark" true...

I wish you could be clearer. Does "dogs bark" mean (a) 'dog' means 'a creature that barks' or (b) 'there is the sound of barking and it is coming from a dog'? I can only repeat my earlier response. If is (a) then no; if it is (b) then yes; some actual dog somewhere has got to bark.

Isn't a round plate shaped like a circle? Of course it's not a perfect circle but it is still has the property of being circular.

It can't have the property of being a geometric circle if it isn't a geometric circle. It might instead have the property of being 'an object that is nearly a circle' i.e. a property like 'roll-ability', but roll-ability is a property only objects can have (it requires three dimensions), not geometrical concepts.

Yes, something may resemble a geometric shape, but resemblance isn't the same thing as being. Your plate resembles a geometric circle...up to the point where it not longer resembles a geometric cirle, being irregular, three dimensional etc. But I might equally say your plate resembles a geometric plane...up to the point where it doesn't. Or that it resembles the number 'one'...but only in respect of its singularity, and so on.

All descriptions of particular objects serve to relate them to general ideas. We might say your plate is 'white', meaning it has something in common with all other white objects, but that isn't a claim that your plate is a colour.

No, it doesn't follow that we can prove God. The abstract concept of a triangle doesn't say that there are in fact triangles, it only says that if something is a triangle then it has such and such properties, so the comparison with the ontological argument seems to me irrelevant.

OK; that is what I say about those circles. That we can conceptualise abstract circles in a geometric sense doesn't mean that there must be real objects that correspond to those abstractions.

I wasn't asking what Kant said, I was asking whether you agree with me or not that mathematics and geometry express synthetic a priori truths about the world, and not simply being true by linguistic convention. And if not then why?

I'm sorry; I have explained that the philosophical term 'synthetic a priori' has more than one meaning - whether we like it or not! The best we can do is make it clear which one we are talking about, which is why I am obliged to keep bringing in people like Kant.

Yet again, I don't know where this 'linguistic convention' idea you keep accusing me of having comes from.

If I must attempt to spell it out, being broadly in sympathy with Kant I think that fundamental concepts (rather more fundamental than maths and geometry) can deserve the description 'synthetic a priori truths', but they are not truths 'about the world', not in the noumena/'things-in-themselves' sense of the world-behind-our-sense-impressions.

But I fear we are just repeating ourselves now. I know you don't like the idea, but I'm sure it would be helpful if you could compare your own ideas to those of Kant, or Schopenhauer say, so that everyone would have a common point of reference.

[Fafner88]

I wish you could be clearer. Does "dogs bark" mean (a) 'dog' means 'a creature that barks' or (b) 'there is the sound of barking and it is coming from a dog'? I can only repeat my earlier response. If is (a) then no; if it is (b) then yes; some actual dog somewhere has got to bark.

I meant there must be dogs which bark, but it doesn't matter which dogs are they (as long as they bark).

To make the point as clear as possible, consider the sentences:

* Barack Obama is bald.

** The president of the US is bald.

The truth of the first one depends on there being one unique person who is bald (namely Obama), while the truth of the second doesn't, because it can be satisfied by an infinite number of different people. So your claim that for a proposition to be about the world it must contain a reference to particular entities is false (but maybe I misunderstood you here and what you meant is the other question about idealized properties).

Yes, something may resemble a geometric shape, but resemblance isn't the same thing as being. Your plate resembles a geometric circle...up to the point where it not longer resembles a geometric cirle, being irregular, three dimensional etc. But I might equally say your plate resembles a geometric plane...up to the point where it doesn't. Or that it resembles the number 'one'...but only in respect of its singularity, and so on.

The point is this: If I draw a very accurate circle I can know a priori without measuring, that the ratio between its radius and circumference will be very close to pi and not 1 or 596, and that's not a fact of physics but a fact of geometry (and thus can be known a priori).

I'm sorry; I have explained that the philosophical term 'synthetic a priori' has more than one meaning - whether we like it or not! The best we can do is make it clear which one we are talking about, which is why I am obliged to keep bringing in people like Kant.

By synthetic a priori I mean this: a proposition that can be known independently of experience which truth can't be derived from linguistic conventions, or it doesn't essentially depend on linguistic conventions.

And to make clear on what exactly I mean by "true by linguistic convention", consider these examples:

* Bachelors are unmarried man.

** Dolphins are mammals.

The first is true simply because we choose to define the word 'bachelor' in a certain way, hence it's true by a convention of language. But "Dolphins are mammals" is something that we have discovered empirically to be true about the world, not because we just choose to define the words that way (I think that people in the past believed that dolphins are fish and not mammals, so the world originally wasn't even associated with the concept of a mammal, unlike the word 'bachelor' that every competent user of language knows that it can't be something other then an unmarried man).

Yet again, I don't know where this 'linguistic convention' idea you keep accusing me of having comes from.

I already quoted you on this-

You can argue that these (maths, geometry) are analytic/tautological; how do you know 'a straight line is the shortest distance between two points'? Because that is the definition of a straight line. It is true 'a priori' but it is 'analytic', not 'synthetic'. It only tells us the meaning of a word, it does not give us knowledge of any actual line that might exist outside our own heads - that must still be gained through experience. (A 'synthetic a priori' would be something that must be true, but is about the world, not just words.)

But I fear we are just repeating ourselves now. I know you don't like the idea, but I'm sure it would be helpful if you could compare your own ideas to those of Kant, or Schopenhauer say, so that everyone would have a common point of reference.

I know very little Kant and nothing at all about Schopenhauer so I'm not going to do that.

[A_Seagull]

[]
I don't think you understood what Kant meant by synthetic a priori judgments or proposition..

I trust you realise that in these forums the first person to resort to ad hominem statements is the loser.

So I win! Yay!

[Wayne92587]

Truth; Our Knowledge of Reality that we Speak of.

Blasphemy; Our Priori Knowledge of Reality that we speak of.

[Belinda]

Fafner wrote:

I doesn't matter if there are lines with no width in reality. Of course geometrical concepts are abstract idealizations, but things in reality can very closely approximate these idealizations. If I draw a very accurate triangle on a page then obviously it will exhibit to a very high degree of approximation the properties that geometry says triangles have (the sum of the angles will be close to 180 degrees, Pythagoras's theorem will be true about it etc.). It will be very strange to say that the properties of actual triangles have nothing to do with the properties of triangles described by geometry. The opposite is true: triangles in reality have these properties because they are the necessary properties of any possible triangle, and these properties can be known a priori to hold of any triangle without examining all possible triangles.

Things in reality "approximate" abstract forms not as approximations of degree but as approximations of kind. Abstractions from real life such as a Euclidean triangle or a Euclidean straight line are unique imaginary concepts. Real life objects are more than concepts, and are not abstracted from reality nor are their attributes unique to each . Except when we pretend for our practical purposes that they possess separable dimensions or other attributes that can be measured.

[Londoner]

Fafner

The truth of the first one depends on there being one unique person who is bald (namely Obama), while the truth of the second doesn't, because it can be satisfied by an infinite number of different people. So your claim that for a proposition to be about the world it must contain a reference to particular entities is false (but maybe I misunderstood you here and what you meant is the other question about idealized properties).

You keep using the word 'particular' and I keep saying 'not particular in the sense of a named individual'.

The point is this: If I draw a very accurate circle I can know a priori without measuring, that the ratio between its radius and circumference will be very close to pi and not 1 or 596, and that's not a fact of physics but a fact of geometry (and thus can be known a priori).

For synthetic a priori we need it to be a fact about physics i.e. the real world. However carefully you draw the circle it won't be.

Me: Yet again, I don't know where this 'linguistic convention' idea you keep accusing me of having comes from.

I already quoted you on this-

And the quote does not contain that phrase.

The quote does not say that terms in geometry are 'linguistic conventions' - it points out they are terms in geometry.

By synthetic a priori I mean this: a proposition that can be known independently of experience which truth can't be derived from linguistic conventions, or it doesn't essentially depend on linguistic conventions.

What does that proposition refer to? Is it about 'the real world', about 'things-in-themselves', about the stuff that originates our experiences?

Or is it a proposition about us? The mental architecture that needs to be in our heads in order for us to think in categories like 'true' or 'false'?

[Fafner88]

You keep using the word 'particular' and I keep saying 'not particular in the sense of a named individual'.

And?

For synthetic a priori we need it to be a fact about physics i.e. the real world. However carefully you draw the circle it won't be.

No, I don't think that all the facts about the world are facts of physics.

And the quote does not contain that phrase.

Saying "because it is the definition" is more or less saying just that.

What does that proposition refer to? Is it about 'the real world', about 'things-in-themselves', about the stuff that originates our experiences?

As I indicted, geometry says true things about the way thing behave in space, on 2D surfaces and 3D bodies. And I don't want to use Kantian terminology.

Or is it a proposition about us? The mental architecture that needs to be in our heads in order for us to think in categories like 'true' or 'false'?

Propositions abut triangles are not about us but about triangles, that's why I insist that they aren't true by virtue of linguistic convention.

[Londoner]

Saying "because it is the definition" is more or less saying just that.

We will just have to differ on that, however if you were mislead originally then by now you should be clear now what actual my position is.

Me: What does that proposition refer to? Is it about 'the real world', about 'things-in-themselves', about the stuff that originates our experiences?

As I indicted, geometry says true things about the way thing behave in space, on 2D surfaces and 3D bodies. And I don't want to use Kantian terminology.

But it doesn't, for the reasons I and other contributors have pointed out.

And besides, when you compare ideas in geometry to 'the way thing behave' our knowledge of 'the way things behave' is derived from experience, however you say:

By synthetic a priori I mean this: a proposition that can be known independently of experience

(my italics)

Propositions abut triangles are not about us but about triangles, that's why I insist that they aren't true by virtue of linguistic convention.

So do you believe that geometric triangles are not ideas but real objects, that exist outside our minds, either in nature or in some metaphysical way? Is this true of other mental concepts, like numbers and God?

(I mention God here because there was once the idea that 'reason', as employed in geometry etc. was a literal 'force' that comes from God. If we are to say that triangles are 'real', then we must ask; 'real...what'? If the answer isn't 'matter' then it must be something like 'spirit', hence the religious aspect.)

[Fafner88]

We will just have to differ on that, however if you were mislead originally then by now you should be clear now what actual my position is.

No, actually your position is not very clear to me.

But it doesn't, for the reasons I and other contributors have pointed out.

Yeah, but I didn't find the arguments very persuasive, because they are based on confusions about language.

And besides, you didn't addressed the argument I put forward for this claim. Do you deny the fact that if I draw an accurate circle then I can know a priori that the ratio between its radius and circumference will be very close to pi? And if you do accept this, then how can you explain this knowledge and reconcile it with your claim that geometry doesn't describe true properties of the physical space?

So do you believe that geometric triangles are not ideas but real objects, that exist outside our minds, either in nature or in some metaphysical way?

The question is ambiguous. In one sense, there are triangles in the world, every kid can draw one on a piece paper. And then there are the geometrical abstract notion of a triangle which is an idealized way of describing the necessary properties of spatial objects, like a triangle on a page. And who knows, maybe it's possible for a perfect triangle to exist in the actual physical space, or at least there could've been a different physics that would allow the existence of such objects.

Is this true of other mental concepts, like numbers and God?

I'm less sure about numbers and I don't believe in God, so let's put these issues aside.