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[Londoner]

So is one example of a-priori knowledge the knowledge that within plane geometry a straight line is the shortest distance between two points? If this and this alone is the sort of thing, i.e. logic and maths, that is a-priori, then this is like the Platonists' claim that there be eternal forms, in my example the axioms of plane geometry would be eternal forms typical of the a-priori.

Yes, mathematics and geometry are obvious candidates for being known a-priori, because it seems that observation isn't required for mathematical or geometrical proofs but only reason. It doesn't prove Platonism though.

Here is my understanding:

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 Kant argues otherwise. He considered that maths provided us with 'synthetic a priori' truths in that they are true independent of experience.

But Kant is coming to it from the angle that the first idea of 'synthetic' is essentially metaphysical. It still references the notion that there is something beyond experience, such that any actual experiences we might have are a suspect copy of that reality. He thinks that philosophical efforts to find a 'synthetic a priori' in that sense are futile.

Instead, philosophy should be concerned with those intuitive concepts (like those of time, and space) that are necessary before we can have any comprehension at all of the world, that these are the true 'synthetic a priori'.

So Kant's 'synthetic a priori' is not the same as other people's. Confusingly, you get philosophers who agree with Kant but deny the possibility of 'synthetic a priori' - because they are referring to the non-Kant version of that phrase.

[Fafner88]

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.)

I don't think that mathematics or geometry is true by definition. "The shortest distance between two points is a straight line" is a true proposition about reality, but one can't gain knowledge about reality by arbitrary defining words. For example, if we choose to define horses as winged animals, it obviously won't make horses winged animals, so why should it be different with mathematics or geometry? (and so a straight line was defined as the shortest distance between two points because it's true, not the other way around).

And a secondly, as Quine convincingly showed in his "Truth by Convention", logic as well can't be simply stipulated tautologies, because one can't stipulate anything intelligibly without first presupposing logical reasoning. So let's take for example modus ponens:

(1) If P then Q (2) P (3) therefore Q

If one stipulates that this rule is true by linguistic convention then one must use something like the following definition:

* If a sentence of the form "If P then Q" is true, and the sentence "P" is true, then the sentence "Q" is true.

But by virtue of which definition this rule is supposed to be true? Perhaps this:

** if a sentence of the form "a sentence of the form "If P then Q" is true" is true, and the sentence of the form "the sentence "P" is true" is true, then a sentence of the form "the sentence "Q" is true" is true.

But again, by virtue of what definition this rule is supposed to be true? You see, we get an infinite regress. One can't stipulate all the rules of one's reasoning, because to understand a stipulation one must already know how to use definitions, which is a procedure that itself guarded by logical rules. So it can't be the case that logical reasoning is valid because we defined it to be so, it is simply valid as a fact (known a priori), and only on the basis of this fact we are able to intelligibly construct logical rules.

[Londoner]

I don't think that mathematics or geometry is true by definition. "The shortest distance between two points is a straight line" is a true proposition about reality, but one can't gain knowledge about reality by arbitrary defining words. For example, if we choose to define horses as winged animals, it obviously won't make horses winged animals, so why should it be different with mathematics or geometry? (and so a straight line was defined as the shortest distance between two points because it's true, not the other way around).

If the meaning of 'horse' was 'winged animal', then surely that would be the meaning of 'horse'? Certainly it wouldn't tell us if such a creature actually existed - a word can exist without anything to refer to.

Similarly, 'a straight line' does not make any claims about reality, unlike 'that straight line'. I could question whether 'that straight line' really was straight, but not whether 'a straight line' was straight, because it is straight by definition - it doesn't claim to exist.

And a secondly, as Quine convincingly showed in his "Truth by Convention", logic as well can't be simply stipulated tautologies, because one can't stipulate anything intelligibly without first presupposing logical reasoning.

That is the Kantian version and surely a different argument. For something to be essential to our logical reasoning it doesn't follow that it must be a true proposition about reality. For example, we might argue that we can only think about the external world if we can name it, using words - but it wouldn't follow that those words we use are a necessary property of those objects. That we have named horses 'horse' tells us a fact about us, not about the nature of horses.

So let's take for example modus ponens:

(1) If P then Q (2) P (3) therefore Q

If one stipulates that this rule is true by linguistic convention then one must use something like the following definition:...

There are two aspects to this. The logical step expressed in (3) but also steps (1) and (2). Those first steps are assumptions; they are falsifiable, they may or may not be true.

If it rains, we will get wet. It is raining. Therefore we will get wet.

But perhaps I have an umbrella? Or perhaps it is actually sunny? To know we will have to fall back on empirical observation, with all the potential unreliability that involves.

If we don't have this contingency, the possibility of the conclusion being false, then this must be because steps (1) and (2) involving P were not falsifiable - in that case they must have been already entailed in the conclusion and it is a tautology. We may as well leave out P, since there is no 'if', and so reduce the whole thing to:

If Q, then Q.

[Fafner88]

If the meaning of 'horse' was 'winged animal', then surely that would be the meaning of 'horse'? Certainly it wouldn't tell us if such a creature actually existed - a word can exist without anything to refer to.

The point was that "the shortest distance between points is a straight line" isn't merely a claim about language (how words are defined) but a claim about reality, and one can't make claims about reality true simply by arbitrary defining words.

Similarly, 'a straight line' does not make any claims about reality, unlike 'that straight line'. I could question whether 'that straight line' really was straight, but not whether 'a straight line' was straight, because it is straight by definition - it doesn't claim to exist.

Of course "a straight line" doesn't make a claim about reality, because it's not a proposition but a term (just like 'water' isn't a claim about reality), but a proposition like "the shortest distance between points is a straight line" does, because it talks about lines points and distances.


The second argument that I have presented is unrelated to the first or to Kant. The first is about mathematics and the second is about logic, and I think you completely misunderstood it.

If it rains, we will get wet. It is raining. Therefore we will get wet.

But perhaps I have an umbrella? Or perhaps it is actually sunny? To know we will have to fall back on empirical observation, with all the potential unreliability that involves.

This seems like a confusion. Modus ponens says that if a conditional is true and the antecedent is true then the conclusion must be true by necessity. In the case you describe the conditional is simply false, so in no way it's an objection to the validity of modus ponens. There's difference between the validity of an argument and the truth of its premises, these are two completely different questions, and logic deals only with the first. Logic (or at least the part of logic that I'm talking about) deals with truth preservation in propositions, not empirical claims about the world.

To recapitulate Quine, he was objecting to the view that logical generalizations are true simply by conventional definitions (like what you tried to claim about geometry), but Quine showed that it's not the case because one can't define anything in the first place without already presupposing the logical rules that one wants to define, and I didn't see how anything that you say deals with this argument.

[Londoner]

Of course "a straight line" doesn't make a claim about reality, because it's not a proposition but a term (just like 'water' isn't a claim about reality), but a proposition like "the shortest distance between points is a straight line" does, because it talks about lines points and distances.

I don't think it does. It what sense do those 'points' and that 'distance' exist except as aspects of the theoretical 'a straight line'? Suppose I asked you 'what is the measurement of that distance?' or 'what are the co-ordinates of those points?' You cannot say, because they are as theoretical as the line.

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.

If not, then I could argue that 'God is omnipotent', is also necessarily true of the concept 'God'. It isn't falsifiable in the same way that your proposition isn't falsifiable; a straight line that wasn't the shortest distance between two points couldn't be a straight line - a God that wasn't omnipotent wouldn't be God. Have I therefore proved God's reality; that God's existence is a 'synthetic a priori' truth?

The answer is; no. You cannot define something into reality; there is still the 'if'. If there is a straight line, or if there is a God, then it must have those characteristics. If.

The second argument that I have presented is unrelated to the first or to Kant. The first is about mathematics and the second is about logic, and I think you completely misunderstood it.

I would agree the second argument was unrelated to the first, but I think it does follow from Kant. But do you not see maths and logic as connected? Can something be true mathmatically, but not logically?

This seems like a confusion. Modus ponens says that if a conditional is true and the antecedent is true then the conclusion must be true by necessity. In the case you describe the conditional is simply false, so in no way it's an objection to the validity of modus ponens. There's difference between the validity of an argument and the truth of its premises, these are two completely different questions, and logic deals only with the first. Logic (or at least the part of logic that I'm talking about) deals with truth preservation in propositions, not empirical claims about the world.

Yes; that was my point. Logic cannot deliver a 'synthetic a priori' in the sense of a necessary truth about the world.

To recapitulate Quine, he was objecting to the view that logical generalizations are true simply by conventional definitions (like what you tried to claim about geometry), but Quine showed that it's not the case because one can't define anything in the first place without already presupposing the logical rules that one wants to define, and I didn't see how anything that you say deals with this argument.

I did not say logic or geometry works by 'convention', in the sense that its rules are arbitrary. If '2 plus 2 equals 4' then necessarily '4 minus 2 equals 2'. As I pointed out, Kant first made the point that our innate understanding of such relationships is a precondition for us to be able to order, and thus make use of, the raw data received by our senses.

However, as we have agreed, the fact that we are obliged to use these concepts does not mean that we are accessing truths about the world. I can use logic and all the rest, yet still be deluded about reality.

(Nor is our understanding of things like logic fixed; we are aware that elements that once seemed necessary for our comprehension of time and space in the age of Newton must be put aside in order to comprehend Einstein or Scrodinger!)

So, to reiterate, we can take the view that we can never overcome Cartesian doubt about our senes and obtain certain knowledge of external reality, so that we should instead apply the phrase 'synthetic a priori' in the sense Kant and Quine use it, as representing necessary cognitive preconditions of all experience. But this is a meaning that relates to us; it abandons the project of finding certain knowledge of things-in-themselves, 'noumena' as opposed to 'phenomena'.

I'm not quite sure of your position. You seem to have the Kant approach, but also seem to suggest that logic etc. can deliver truths about 'things-in-themselves'.

[Fafner88]

I don't think it does. It what sense do those 'points' and that 'distance' exist except as aspects of the theoretical 'a straight line'? Suppose I asked you 'what is the measurement of that distance?' or 'what are the co-ordinates of those points?' You cannot say, because they are as theoretical as the line.

How can you say that points and lines don't exist if one can draw them on a piece of paper? I don't quite understand what you mean by 'theoretical'. Electrons are 'theoretical' and yet they exist and we can talk about them. Points and lines in geometry are indeed abstractions, but they do describe genuine properties of the actual physical space we live in (ignoring the fact that it's non-euclidean according to modern physics).

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.

What about "dogs bark", can you locate the coordinates of the dogs which this sentence is about? It's plainly obvious that one can make general claims about reality without referring to particular objects or locations. Of course when we say that "the shortest distance between points is a straight line" no one has any particular line or points in mind, but it doesn't mean that we are not talking about reality (and it's not clear to me what is your proposal, that geometry is about language?).

The answer is; no. You cannot define something into reality; there is still the 'if'. If there is a straight line, or if there is a God, then it must have those characteristics. If.

Duh, this was exactly my point, mathematical and geometrical claims are substantive claims about reality, and since one can't simply create such truths by defining words, it means that mathematics and geometry can't be true by convention or definition like on your original proposal (synthetic if you like).

I would agree the second argument was unrelated to the first, but I think it does follow from Kant.

Let's forget about Kant and just concentrate on the arguments.

But do you not see maths and logic as connected?

Yes perhaps, I would say that it's impossible to do mathematics without presupposing logic, for example if you try to prove something (you have to assume the law of contradiction etc.).

Can something be true mathmatically, but not logically?

I don't understand this question.

Yes; that was my point. Logic cannot deliver a 'synthetic a priori' in the sense of a necessary truth about the world.

I never said that it could. I simply claimed that deductive rules of inference can't be true simply by virtue of definition.

I did not say logic or geometry works by 'convention', in the sense that its rules are arbitrary.

You did say that. You said that a straight line is the shortest distance between two points because it's defined that way. And if the definitions are not arbitrary then where on your view, their truth is derived from if not from convention?

If '2 plus 2 equals 4' then necessarily '4 minus 2 equals 2'. As I pointed out, Kant first made the point that our innate understanding of such relationships is a precondition for us to be able to order, and thus make use of, the raw data received by our senses.

It doesn't matter what Kant said, I just want to see the arguments. I don't want to turn this into a Kant exegeses.

However, as we have agreed, the fact that we are obliged to use these concepts does not mean that we are accessing truths about the world. I can use logic and all the rest, yet still be deluded about reality.

No, we didn't agreed on this. I don't understand what else the sentence "a straight line is the shortest distance between two points" is supposed to be about if not about lines and points? The question was, by virtue of what a sentence like this is true? Your original proposal, as I understood it, was that it's simply a matter of definition or analiticity, and I ask what are your reasons for thinking that.

So, to reiterate, we can take the view that we can never overcome Cartesian doubt about our senes and obtain certain knowledge of external reality, so that we should instead apply the phrase 'synthetic a priori' in the sense Kant and Quine use it, as representing necessary cognitive preconditions of all experience. But this is a meaning that relates to us; it abandons the project of finding certain knowledge of things-in-themselves, 'noumena' as opposed to 'phenomena'.

I don't see how this topic is related to Cartesian doubt. Many sceptics will concede that we may know a priori truths about mathematics and geometry, they only deny knowledge about empirical facts (and of course I never said that mathematics or geometry are known empirically).

[Belinda]

Fafner wrote:

How can you say that points and lines don't exist if one can draw them on a piece of paper? I don't quite understand what you mean by 'theoretical'. Electrons are 'theoretical' and yet they exist and we can talk about them. Points and lines in geometry are indeed abstractions, but they do describe genuine properties of the actual physical space we live in (ignoring the fact that it's non-euclidean according to modern physics).

You can say that mathematical points and lines have no empirical existence. In plane geometry points and lines exist either as abstractions from empirical facts, or they exist in the realm of eternal, not empirical, forms. Plato and Pythagoras obviously believed that eternal forms were eternal facts . It is usually contended nowadays that points and lines are useful models that are abstracted from empirical reality. Indeed it is credible that plane geometry has its historical beginnings when the fertile lands of the Nile margins were being apportioned to farmers and owners.

When we draw points and lines we can only represent the ideas. It is impossible to draw an actual point or an actual straight line because those, like all of maths and logic, have no empirical existence.

Electrons, (I checked with a professional physicist about one year ago) are not like purely abstract measurements because electrons do exist empirically as well as being mathematically verified.

Fafner wrote:

You did say that. You said that a straight line is the shortest distance between two points because it's defined that way. And if the definitions are not arbitrary then where on your view, their truth is derived from if not from convention?

That a Euclidean point has no dimensions is an axiom, a self evident truth (Kantian synthetic a priori if you like). A straight line is defined so as to follow logically from axiomatic two points. Euclidean(plane) geometry is a system for measuring space : it does not cause space or objects in space to exist. Any measuring system, whether it measures time or any other dimension, is arbitrary as an entire system but any logically valid system's details are not arbitrary because the details follow logically from the initial axioms.

[Londoner]

Fafner

What about "dogs bark", can you locate the coordinates of the dogs which this sentence is about? It's plainly obvious that one can make general claims about reality without referring to particular objects or locations.

Again, I disagree. Either this is a statement about the meaning of 'dog' ('an animal that barks') or alternatively it describes an actual event that may or may not have occured. Whether or not it has occured is something we would have to determine empirically.

Duh, this was exactly my point, mathematical and geometrical claims are substantive claims about reality, and since one can't simply create such truths by defining words, it means that mathematics and geometry can't be true by convention or definition like on your original proposal (synthetic if you like).

I do not understand you, or rather perhaps you do not understand me. 'Synthetic' does not mean 'true by convention'. The synthesis in 'synthetic' is that their truth depends on us knowing the meaning of the words and also something about the world (as opposed to 'analytic' where just the meaning of the word is sufficient).

'Dogs bark' could be either; it could be just about the meaning of the word 'dog' or it could also be a claim about the world. That these are not the same thing was what I was trying to illustrate with my earlier example of God. We could accept 'God is omnipotent' might be true analytically, but we would not accept that it was therefore also true synthetically i.e true about the world; that God was therefore a thing that exists.

Me: I did not say logic or geometry works by 'convention', in the sense that its rules are arbitrary.

You did say that. You said that a straight line is the shortest distance between two points because it's defined that way. And if the definitions are not arbitrary then where on your view, their truth is derived from if not from convention?

They are derived from the context, in this case the context of Euclidean geometry. It is however possible to have non-Euclidean geometry, in which case we will have a different set of definitions. In another area, such as that dealing with material objects, we instead understand events in terms of 'cause and effect'. And again, we understand the actions of people in terms of psychological forces; 'will' etc. We think in these various areas using particular tools because they provide us with a coherent and useful model with which to understand them.

(These contexts don't mix. A Euclidean explanation doesn't work in non-Euclidean geometry. A psychological explanation isn't valid to explain the behaviour of matter and so on. (See Schopenhauer & co.))

So their being non-arbitrary derives from their functionality. If our understanding changes and they cease to function, or we see a problem in a new context, we will readily discard one set of 'definitive tools' and apply another.

[Fafner88]

Again, I disagree. Either this is a statement about the meaning of 'dog' ('an animal that barks') or alternatively it describes an actual event that may or may not have occured. Whether or not it has occured is something we would have to determine empirically.

But why? Do you have any arguments? This is such an implausible claim. When I say "there are cats in Australia" I don't have in mind any particular cat (or definition??) because I never been to Australia and I don't know any cats from there, and yet it's a completely intelligible sentence which is also true.

Duh, this was exactly my point, mathematical and geometrical claims are substantive claims about reality, and since one can't simply create such truths by defining words, it means that mathematics and geometry can't be true by convention or definition like on your original proposal (synthetic if you like).

I do not understand you, or rather perhaps you do not understand me. 'Synthetic' does not mean 'true by convention'. The synthesis in 'synthetic' is that their truth depends on us knowing the meaning of the words and also something about the world (as opposed to 'analytic' where just the meaning of the word is sufficient).

Yes, I didn't mean that synthetic is true by virtue of definition, because that's analytic, I put the brackets in the wrong place. I meant to say that on my view mathematics and geometry are synthetic, and if I understood you correctly you claim that they are analytic, that's the main disagreement as far as I can see (but we agree that in either case they must be known a priori).

We could accept 'God is omnipotent' might be true analytically, but we would not accept that it was therefore also true synthetically i.e true about the world; that God was therefore a thing that exists.

But do you agree that synthetic claims can't become true simply by definition?

They are derived from the context, in this case the context of Euclidean geometry. It is however possible to have non-Euclidean geometry, in which case we will have a different set of definitions.

This is perfectly fine, indeed euclidean and non-euclidian geometries are different, but it doesn't follow that they are a mere linguistic definitions. They both say true things, but about two different kinds of space. whatever one thinks about non-euclidian geometry, it's still true that in a euclidian space the shortest distance between two points is a straight line, and this truth doesn't simply depend on our decision. Every euclidian space will have this property, whether there is such a space in reality or not.

So their being non-arbitrary derives from their functionality. If our understanding changes and they cease to function, or we see a problem in a new context, we will readily discard one set of 'definitive tools' and apply another.

I don't see how being 'dependent on context' entails being true by linguistic convention. To take a trivial example, if I say "it's raining", the truth of this sentence depends on the context, namely my location, but this of course doesn't make the fact itself that it's raining (where I stand) dependent on linguistic conventions.

[Londoner]

When I say "there are cats in Australia" I don't have in mind any particular cat (or definition??) because I never been to Australia and I don't know any cats from there, and yet it's a completely intelligible sentence which is also true.

Its truth would be determined empirically by going to a location 'Australia' and looking for cats. I agree, not a particular cat ('Felix'), but that particular type of animal. We have both an object and a location, so as an example it contrasts with what you had written earlier:

What about "dogs bark", can you locate the coordinates of the dogs which this sentence is about? It's plainly obvious that one can make general claims about reality without referring to particular objects or locations.

But do you agree that synthetic claims can't become true simply by definition?

I do; I thought I had made it plain that 'synthetic' includes a claim about reality.

This is perfectly fine, indeed euclidean and non-euclidian geometries are different, but it doesn't follow that they are a mere linguistic definitions.

This 'mere linguistic' bit is your interpretation of what I wrote.

Euclidean terms accurately describe Euclidean geometry. The point is that Euclidean geometry doesn't describe everything - what is true within Euclidean geometry isn't necessarily true outside it. So we cannot claim that because its concepts, like 'a straight line is the shortest distance...' are necessarily true within that context, they are also necessarily true of external reality.

I don't see how being 'dependent on context' entails being true by linguistic convention. To take a trivial example, if I say "it's raining", the truth of this sentence depends on the context, namely my location, but this of course doesn't make the fact itself that it's raining (where I stand) dependent on linguistic conventions.

Again, this 'liguistic convention' description is yours, not mine.

To take up your example, the context of "it's raining" is not your own physical location, but the context of the discussion of which that remark was a part. If it is understood as an empirical claim, then its truth or falsity is tested by sensory observation; if others can also feel the rain-drops we accept it as evidence it is true.

But even such a prosaic example could have other contexts. If we were discussing our state of mind it might mean "it's raining (in my heart)" in which case others could not verify the truth by direct sensory evidence. OK, that is a bit fanciful in this particular instance, but it is not always the case. What about the philosophical claim that "a 'thing-in-itself' is the cause of our experiences"? What type of reasoning/experience is the appropriate means of verifying that assertion?

[Fafner88]

Its truth would be determined empirically by going to a location 'Australia' and looking for cats. I agree, not a particular cat ('Felix'), but that particular type of animal.

Of course, it's an empirical questions whether there are cats in Australia.

so as an example it contrasts with what you had written earlier:

What about "dogs bark", can you locate the coordinates of the dogs which this sentence is about? It's plainly obvious that one can make general claims about reality without referring to particular objects or locations.

No, it's not that different. Only that "dogs bark" is ambiguous and so perhaps it wasn't the best example. It can mean either "there are some dogs who bark" or "all dogs bark", but notice that on both readings, we don't have to refer to any particular dog for the sentence to make sense, just as in the example of the cats in Australia.

To go back to the original point: you said that geometrical propositions can't be about reality because one can't locate abstract lines or points in space and time. But as my last example shows, this by itself doesn't mean that the proposition can't be synthetic. There's no particular line which the concept of 'straight line' in geometry is about, just as there's no particular dog about whom we talk when we say "dogs bark", but it doesn't mean that there are no lines in the world that geometry can say about them true things. To put it in other words: "dogs bark" is true if there are actual dogs that bark (in space and time etc.), but my point is that the meaning of "dogs bark" doesn't depend essentially on the existence of any particular dog (and what a sentence mean and whether it is true are of course two different questions).

This 'mere linguistic' bit is your interpretation of what I wrote.

So where you don't agree with me?

To quote you original post:

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.)

I understood you as proposing something on the lines of what Hume and later the logical positivists claimed, namely that mathematics and geometry are true simply by matter of linguistic convention (which is also supposed to explain why they are necessary). So if you don't agree with this view, what on your view the status of geometrical and mathematical truths is? Are they true because the way of the world is or because of linguistic conventions (like "bachelors are unmarried man"), or something else?

[Spectrum]

I understood you as proposing something on the lines of what Hume and later the logical positivists claimed, namely that mathematics and geometry are true simply by matter of linguistic convention (which is also supposed to explain why they are necessary). So if you don't agree with this view, what on your view the status of geometrical and mathematical truths is? Are they true because the way of the world is or because of linguistic conventions (like "bachelors are unmarried man"), or something else?

I mentioned earlier, in the case of the term 'a priori' it is essential to agree on what the term is to be used for before proceeding to discuss the related issue. IMO, it is because this was not done, that you and Londoner are unable to reconcile each other points.

Noted your use of a priori is more traditional and confined to those used by the analytic philosopher while Londoner is using it in Kant's perspective (my preference as well). It was Kant who made the term 'a priori' popular (or controversial) in modern philosophy using his own specific definition of what is 'a priori'. Kant agreed with the traditional view of 'a priori' but he extended the meaning with more precision by adding the qualifications of 'necessary' and 'universal'. In addition there are mixed and absolute/pure a priori elements.

What is critical for Kant in his use of a priori is the justifications and conclusions that he can deduced from his interlinked-System of knowledge, via the term 'synthetic a priori proposition,' e.g. God and its metaphysical elements are illusory, absolute freedom is the ground of Morality, etc.

Mathematical axioms are pure a priori knowledge in the pure aspects but they are justified and true in their applied aspects. Their possibility guarantees their truth. There are no serious issues in terms a priori (analytic or synthetic) with Mathematics or Science as the a priori grounds can be justified by their possibility and empirical evidence.

The big issue with a priori is the synthetic a priori proposition in Metaphysics culminating with the mother (or rather father) of all issues, i.e. God exists?

[A_Seagull]

[ we should instead apply the phrase 'synthetic a priori' in the sense Kant and Quine use it, as representing necessary cognitive preconditions of all experience. .

I would agree that there are fundamentals that are preconditions for cognitive experience. But to simply label them as 'synthetic a priori' really achieves nothing. In fact it is counter productive as it implies that those fundamentals cannot be understood in any simpler form. It acts as a block to investigation.

Further, IMO those fundamentals take the form of an algorithm which is able to extract patterns from strings of data. It requires no preconception of any form of the world. Further details are to be found in 'The Pattern paradigm'.

[Spectrum]

[ we should instead apply the phrase 'synthetic a priori' in the sense Kant and Quine use it, as representing necessary cognitive preconditions of all experience. .

I would agree that there are fundamentals that are preconditions for cognitive experience. But to simply label them as 'synthetic a priori' really achieves nothing. In fact it is counter productive as it implies that those fundamentals cannot be understood in any simpler form. It acts as a block to investigation.

Further, IMO those fundamentals take the form of an algorithm which is able to extract patterns from strings of data. It requires no preconception of any form of the world. Further details are to be found in 'The Pattern paradigm'.

I don't think you understood what Kant meant by synthetic a priori judgments or proposition.
Kant acknowledge synthetic a priori judgments are possible in natural science (together with other knowledge) and one can use science to study any empirical preconditions for cognitive experience. The test is whether the resultant can be tested and proven within science.

Kant a priori elements are different from the 'empirical' algorithm you speak of.
Kant's categories [one set] are fundamental a priori Pure Concepts of Understanding not of empirical patterns you speak off.
One fundamental category is that of Cause and Effect. Surely your pattern paradigm must be subjected under the law of cause and effect, i.e. what is the cause of the patterns, etc. and subsequently how are you to deal away with the issue of a first cause.

[Londoner]

Faffner

No, it's not that different. Only that "dogs bark" is ambiguous and so perhaps it wasn't the best example. It can mean either "there are some dogs who bark" or "all dogs bark", but notice that on both readings, we don't have to refer to any particular dog for the sentence to make sense, just as in the example of the cats in Australia.

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.

To go back to the original point: you said that geometrical propositions can't be about reality because one can't locate abstract lines or points in space and time. But as my last example shows, this by itself doesn't mean that the proposition can't be synthetic. There's no particular line which the concept of 'straight line' in geometry is about, just as there's no particular dog about whom we talk when we say "dogs bark", but it doesn't mean that there are no lines in the world that geometry can say about them true things. To put it in other words: "dogs bark" is true if there are actual dogs that bark (in space and time etc.), but my point is that the meaning of "dogs bark" doesn't depend essentially on the existence of any particular dog (and what a sentence mean and whether it is true are of course two different questions).

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.

It is like numbers; in the world of things you can have an apple and add the description 'one' and say 'one apple', but you can't have a descriptive 'one' that isn't attached to an object. 'One apple' minus an apple doesn't leave the 'one'!

Once again, I agree the truth of 'dogs bark' doesn't depend on the dog being a known individual ('Fido'). But it does depend on the existence and actions of a real thing that corresponds to our idea of a dog. We can't claim 'dogs bark' unless we also claim 'dogs exist'.

It is true that someone might have an unusual notion of 'dog'; e.g. 'dog' might mean 'fictional beast that barks'. But then 'dogs bark' would not be a synthetic claim because such claims also need to include the analytic component i.e. a specific meaning of the word 'dog'. And if that meaning was the 'fictional beast' version, then 'dog's bark' would only be analytic; there would be no claim about reality - its truth would only only hang on that definition.

I understood you as proposing something on the lines of what Hume and later the logical positivists claimed, namely that mathematics and geometry are true simply by matter of linguistic convention (which is also supposed to explain why they are necessary). So if you don't agree with this view, what on your view the status of geometrical and mathematical truths is? Are they true because the way of the world is or because of linguistic conventions (like "bachelors are unmarried man"), or something else?

I think they are 'true by necessity' (!), in that unless we have such ways of ordering our thoughts/sense impressions then we cannot operate.

But it does not follow that because they are necessary for us it proves that they are true of the world beyond us. Crudely; it might be necessary for me to work on the assumption that things I see with my eyes are external objects, not figments of my imagination, but my need isn't evidence that this is true in a metaphysical sense.

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.