Philosophy Discussion Forums | A Humans-Only Club for Open-Minded Discussion & Debate

[Fafner88]

It is a valid argument, but however valid, for the conclusion of an argument to be true the premises must be true. One of your premises is an assumption; if the conclusion was meant to be a truth 'synthetic a priori' then it can't depend on an assumption.

Why not? You forget the goal of the argument, it is supposed to show that we can know empirical facts about the world using a priori premises (namely premise 1), and if we can do that, then these a priori premises are very likely to be synthetic (but this is the second stage of the argument as it were). And hence it doesn't matter if you call the conclusion (3) synthetic a priori knowledge or not, the point is that it's a knowledge about an object in the world which was achieved using an a priori premise. And now if you agree that the argument is valid, then what's wrong with it?

So do you argue that geometry is synthetic? In that case I would again point out that its claims aren't true, since no geometric shapes can exist as objects.

Well no, propositions in geometry are not about physical objects, nobody thinks that (including myself). Geometry deals with abstract concepts which don't essentially depend on the physical reality (there needn't be any actual triangles for Phytagorase's theorem to be true). But what's also true on my view is that geometry allows us to know different properties that physical objects must exhibit, and this fact is very difficult to explain unless you suppose that geometry is synthetic (but this is not to say that it's empirical like science).

By contrast, my reasons for thinking that theorems in geometry are analytic is that they can be proved by reason alone - a theorem is that which has been proved to be true using facts that were already known. We can use algebra to prove Pythagoras in which there is no mention of specific measurements.

This argument also begs the question. Saying that something must be analytic because it's a priori is not a very good argument against someone who thinks that not all a priori knowledge is analytic. You just assume the thing which is disputed...

The problem is skepticism about knowledge in general. The search for a synthetic a priori is the search for an answer to the question: How can we know anything for sure about any thing? If you assume that X both is a thing and it is this sort of a thing you have assumed what you need to prove.

Well no, it's only in your mind a question about scepticism while in the philosophical literature the two questions are kept apart, and for a good reason. That's not a very interesting argument to say that since we can't know anything then there is no synthetic a priori knowledge. As I said, my argument is not aimed at sceptics, and since most people are not sceptics, then this argument can still be interesting because knowing that something a triangle is a trivial piece of knowledge that no reasonable person will dispute, and it's a good thing to have an argument with uncontentious premises (while your premises are extremely contentious e.g. that there are no triangles in reality).

This is the same argument as before, except that you have run two things together in the first part. There is 'if P' and separately 'if P is a triangle'. Yes; if P is a triangle then it will have the properties of a triangle. But that doesn't deal with the 'if' in 'if P exists'.

?

You would just have to accept that when anyone else refers to a 'cat', they refer to something with a tail. The proposition 'Cats have tails' would be analytic; the predicate is contained in the subject.

Wait a second, if you cut a cat's tail then it cease to be a cat? Are you serious?

Anyway, this question is unrelated to the main topic, because the example of the cat was only meant to show that one can't use an argument of the sort that I used to know ordinary empirical propositions, while you can use it to know things about triangles and the like, which is in my opinion quite an impressive fact (and this goes back to my first example that one can't make it the case that horses have wing simply by redefining words, and it seems to me wrong to say that when Pythagoras discovered his theorem he simply played with words, because he clearly made a significant discovery about reality).

[Gmanley7]

There are non-empirical ways to come to truths, (like thinking seriously and honestly about an issue, like philosophers do) however you would never know for sure whether your conclusion is really true until you subject it to experience: say after much thinking ( reasoning a alone) you figured that rid the planet of all evil, poor and uneducated people through violent means was the way to give the world a quick fresh start, solve the problem of poverty and evil and guarantee that future generations of humans will live peacefully and happy. Even if no one could point any hole in your reasoning, only putting this conclusion into practice in the real  world would say whether you were right to have reasoned so.( But we know for sure that this one doesnt work, because apparently the God of the Bible tried it with the deluge and look where we are now). Thus experience is the ultimate arbiter of knowlege. It's sometimes possible to prove logically that that box you see in the forest must have gold in it, whereas your friend next to you can logically demonstrate it can only have a dead cat in it. you and your friend may argue for ever about what's in the boxe, but only opening the boxe ( which is the experiment ) can tell if anyone is right. A priori reasoning, thus apriori knowledge relies strictly on logic, but the laws of logic are all derived or informed by experience. for instance if you lived before the 1900s, then it is absurd that anything can be at different places at the same time. But if you  are reasoning now, that somethings (particles) can be at many places at the same time should be part of what is logical to you. No matter how much you sat on an armchair and think about it , a priori reasoning would never lead you to  this conclusion, or to the possibility of time travel, or to the possibility of uncaused material events.

[Londoner]

Fafner

And hence it doesn't matter if you call the conclusion (3) synthetic a priori knowledge or not, the point is that it's a knowledge about an object in the world which was achieved using an a priori premise. And now if you agree that the argument is valid, then what's wrong with it?

You understand that the 'argument' is just the logic bit? 'If women have tails; and if trees are women; then trees have tails' is a 'valid argument'.

But back to the main point: you have an 'a priori' premise about Euclidean geometry, plus an assumption that we have knowledge of a real world object. You can't glue those two together to make one 'a priori'; they are independent - the first part may be true but the second may be false.

One plus one equals two. Suppose there are 'two apples'. Have we thus proved that 'two apples' must exist, because that description includes a mathematical idea derived purely from reason?

Saying that something must be analytic because it's a priori is not a very good argument against someone who thinks that not all a priori knowledge is analytic. You just assume the thing which is disputed...

I do not say 'a priori'. I say that in the case of geometry, we must assume certain postulates. The truth of any theorem ultimately rests on their truth; it is 'deduced' from them. You say;

when Pythagoras discovered his theorem he simply played with words, because he clearly made a significant discovery about reality.

(I'll ignore your choice to use 'played with words' to gloss 'analytic'!)

If Pythagoras had made a discovery about reality, then what he showed would be true in all circumstances. Yes, his theorem follows necessarily from his postulates. But as others have pointed out, if you change either the postulates or conditions (including the very unreal one of 'reality' consisting of two-dimensional space) his theorem no longer works.

Rather than keep chasing this around, can I again signal the step philosophers have previously taken to try and get round the problem that you cannot have that sure knowledge of a real world object required for your argument?

They have noted that the reasoning about geometry etc. that you consider 'a priori' is something that takes place within our own heads. (This is different to Kant, who I know I mustn't mention.) So instead, why not treat 'our own heads' as the 'real thing'? If you are looking for an object that both contains geometrical reasoning and has physicality - what about yourself? Londoner can dispute you can know that triangle exists, but he couldn't dispute that you know you exist!

I won't try to take that line further; I just point it out as the reason why mainstream philosophy regarding a 'synthetic a priori' has tended to revolve around the nature of consciousness and knowledge.

[Fafner88]

You understand that the 'argument' is just the logic bit? 'If women have tails; and if trees are women; then trees have tails' is a 'valid argument'.

The premises of this argument are neither true nor can be known a priori so I don't see how it's relevant.

But back to the main point: you have an 'a priori' premise about Euclidean geometry, plus an assumption that we have knowledge of a real world object. You can't glue those two together to make one 'a priori'; they are independent - the first part may be true but the second may be false.

what do you mean by 'glue together'? The aim of the argument was to show that a priori knowledge about geometry allows us to know things about the physical world, and if it's the case then geometry is likely to be synthetic (though of course this further claims needs some more defense).

One plus one equals two. Suppose there are 'two apples'. Have we thus proved that 'two apples' must exist, because that description includes a mathematical idea derived purely from reason?

Of course not, but how it follows from my argument that anything 'must' exist? I don't see the point.

I do not say 'a priori'. I say that in the case of geometry, we must assume certain postulates. The truth of any theorem ultimately rests on their truth; it is 'deduced' from them.

This is true, but the postulates are not arbitrary defined the way "bachelors are unmarried" is, and so it is hard to see how geometry and mathematics can be such powerful tools for describing reality (physics etc.) if they are analytic the same way as "bachelors are unmarried" (and the claim is that knowing that "bachelors are unmarried" doesn't allow you to make significant discoveries about the world).

when Pythagoras discovered his theorem he simply played with words, because he clearly made a significant discovery about reality. (I'll ignore your choice to use 'played with words' to gloss 'analytic'!)

Well let's say that Pythagoras didn't simply define his words to make the theorem true by definition, or something of that sort. I mean if what Pythagoras has discovered is analytic, then in what sense it is a 'discovery'? (learning that "bachelors are unmarried" doesn't seem like a discovery about anything, it's simply a definition)

If Pythagoras had made a discovery about reality, then what he showed would be true in all circumstances. Yes, his theorem follows necessarily from his postulates. But as others have pointed out, if you change either the postulates or conditions (including the very unreal one of 'reality' consisting of two-dimensional space) his theorem no longer works.

The theorem will always work in one sense, because it's an abstraction which doesn't depend on the physical world, but of course I don't deny that you can use the theorem only to describe euclidean space or something that approximates euclidean space.

Rather than keep chasing this around, can I again signal the step philosophers have previously taken to try and get round the problem that you cannot have that sure knowledge of a real world object required for your argument?

I don't believe that one can "get round" the problem of knowledge of the external world, either one can know the things outside oneself or one can't, at least this is my view (and of many contemporary philosophers).

They have noted that the reasoning about geometry etc. that you consider 'a priori' is something that takes place within our own heads. (This is different to Kant, who I know I mustn't mention.) So instead, why not treat 'our own heads' as the 'real thing'? If you are looking for an object that both contains geometrical reasoning and has physicality - what about yourself? Londoner can dispute you can know that triangle exists, but he couldn't dispute that you know you exist!

Saying that the 'real things' are just what's insides one's own head doesn't seem to me like a very satisfactory 'solution'. And if you are a sceptic about the outside world then you are not allowed to assume that you are a physical being (or that you have a head).

[Belinda]

Fafner wrote:

Londoner wrote:

You understand that the 'argument' is just the logic bit? 'If women have tails; and if trees are women; then trees have tails' is a 'valid argument'.

The premises of this argument are neither true nor can be known a priori so I don't see how it's relevant.

Londoner illustrated a valid argument, the conclusion of which is implicit in the premises. That one or more premises are false does not invalidate the argument.

Similarly the axiomatic premises of Euclidean geometry imply every truth that is deduced from them. If some farmer measured the side of a field incorrectly the argument for discovering the area of the field is valid although the premise regarding the length of the fence be incorrect. The fault would lie not in the validity of the argument but in the empirical measurement.

[Londoner]

This is true, but the postulates are not arbitrary defined the way "bachelors are unmarried" is, and so it is hard to see how geometry and mathematics can be such powerful tools for describing reality (physics etc.) if they are analytic the same way as "bachelors are unmarried" (and the claim is that knowing that "bachelors are unmarried" doesn't allow you to make significant discoveries about the world).

It does not help if you keep glossing my arguments using words like 'arbitrary' which I have not used.

The theorem will always work in one sense, because it's an abstraction which doesn't depend on the physical world, but of course I don't deny that you can use the theorem only to describe euclidean space or something that approximates euclidean space.

It must be entirely Euclidean space or the geometry doesn't work. Euclidean space does not exist, anywhere, ever.

Well let's say that Pythagoras didn't simply define his words to make the theorem true by definition, or something of that sort. I mean if what Pythagoras has discovered is analytic, then in what sense it is a 'discovery'? (learning that "bachelors are unmarried" doesn't seem like a discovery about anything, it's simply a definition)

Again, the 'simply define' bit is a gloss that doesn't reflect anything I have argued.

As I have said, I don't think it is a discovery in the sense of a new fact, like 'there exists a great continent over the ocean to the west!'. Given the various postulates, could Pythagoras' theorem not be true? Then what else could it be but an expression of those postulates?

I don't believe that one can "get round" the problem of knowledge of the external world, either one can know the things outside oneself or one can't, at least this is my view (and of many contemporary philosophers).

If you say so. But I seem to recall there has been, and continues to be, rather a lot of discussion about what 'know' in that sentence might mean.

[Fafner88]

It does not help if you keep glossing my arguments using words like 'arbitrary' which I have not used.

So it would be helpful if you could explain what in general do you think that the synthetic/analytic distinction is supposed to be about. My understanding of 'analytic' is just the traditional way philosophers usually understood it, as something which is true by virtue of an arbitrary linguistic convention, and "bachelors are unmarried" being the paradigmatic example of an analytic truth. So I don't know what else could you mean by 'analytic' if not this.

It must be entirely Euclidean space or the geometry doesn't work. Euclidean space does not exist, anywhere, ever.

But you can't deny that physical space does behave in a euclidean fashion on the small scales of ordinary sized objects (but this is really an inessential point, because some type of non euclidean geometry is strictly true about physical space as relativity says, so we can replace euclidean geometry with something else).

As I have said, I don't think it is a discovery in the sense of a new fact, like 'there exists a great continent over the ocean to the west!'. Given the various postulates, could Pythagoras' theorem not be true? Then what else could it be but an expression of those postulates?

But the discovery that E=MC^2 isn't like that either, but it doesn't mean that relativity wasn't a discovery of new facts about the world.

[Belinda]

Analysing is separating something into parts such as separating some specific sentence into its component clauses, or separating out the elements in a compound or mixture. Analysing therefor informs what something is made of , or its measurements, according to some accepted method, but does not add any new knowledge about the world.

Synthesising means adding to ourselves new knowledge about the world; or it might mean building a new substance . The problem with the truth of a synthetic judgement is that it can never be 100% true, as can an analytic judgement.

The problem with an analytic judgement is that it is not entirely new, but is a glorified tautology.

[Daviddunn]

It doesn't quite follow, because the reverse also must be true, but is Pythagoras' theorem contained in the notion of a right angle triangle? I think not because as I said, someone can know what a right angle triangle is without knowing anything about the theorem, while one can't know what 'bachelor' means without also knowing that all bachelors are unmarried man (because then he simply doesn't understand the word). Perhaps it doesn't show that Pythagoras' theorem isn't analytic, but frankly it's quite difficult do see how it can be analytical while competent users of language can still fail to see the truth of the theorem. If indeed Phytagoras' theorem is true by virtue of the meaning of the words, how can someone understand the words while not knowing the theorem?

I understand your point now. So instead of:

If x is a right angled triangle, then the square of the hypothenuse of x= the square of the opposite of x + the square of the adjacent of x.

you wrote:

If x is a right angled triangle then pythagoras theorem is true of x.

I agree, it is synthetic.

[Londoner]

Fafner

So it would be helpful if you could explain what in general do you think that the synthetic/analytic distinction is supposed to be about. My understanding of 'analytic' is just the traditional way philosophers usually understood it, as something which is true by virtue of an arbitrary linguistic convention, and "bachelors are unmarried" being the paradigmatic example of an analytic truth. So I don't know what else could you mean by 'analytic' if not this.

The word 'bachelor' refers to a concept. True, we might have instead chosen the word 'caelibem' (Latin) to refer to that concept, and provided everyone understood it then that would do, but what we are interested in is the concept, not the sign for that concept.'A bachelor is an unmarried man' is an accurate description of the concept 'bachelor'.

Belinda describes 'analytic' as a glorified tautology, but that doesn't imply it is so obvious. When you ask me what I mean by 'analytic'; you are asking me to give the equivalent of 'a bachelor is an unmarried man' answer...although in this case it is far more difficult. Far from being arbitrary, the meaning of 'analytic' is a subject of contention amongst philosophers.

So, although a dictionary of philosophy would say 'Analytic propositions are propostions that are true by virtue of their meaning', that doesn't tell us whether a proposition falls into that category - and if it does, exactly what that meaning would be. Indeed, isn't this expressed by the reflex response of philosophers to any question i.e. 'It depends what you mean by...'?

But you can't deny that physical space does behave in a euclidean fashion on the small scales of ordinary sized objects (but this is really an inessential point, because some type of non euclidean geometry is strictly true about physical space as relativity says, so we can replace euclidean geometry with something else).

I disagree. If we were to try to create a form of geometry that was true of real objects in the same way that Euclidean geometry is true of two dimensional ones, we would first have to have the equivalent of his postulates. These postulates would have to be (necessarily true) about real objects.

But our knowledge of real objects is unsure, so we cannot derive those postulates from experience. Any postulates we did assert would have to be derived from an understanding of the universe - but that understanding was supposed to be what we derived from those postulates, not the other way round. (e.g. we have to first guess the universe is what it seems to be, before we could claim as a postulate that our senses provide accurate knowledge)

But the discovery that E=MC^2 isn't like that either, but it doesn't mean that relativity wasn't a discovery of new facts about the world.

Unlike geometry, that formula is ultimately based on science; unlike a theorem its truth or otherwise would be determined by empirical evidence.

This isn't to claim that empirical evidence is conclusive, however the conventions of science accept as an assumption that they are, and also that certain sorts of reasoning are valid.

Earlier in this exchange I mentioned the idea that the truth of statements are understood in their context. What verifies a proposition in science isn't the same as what verifies one in geometry (or psychology etc.) Similarly, the assumptions required for one are different to those required for another. That is why you cannot mix them. For example 'I dream of monsters' is evidence of a psychological state, it may be a fact, but it cannot be used as evidence of a scientific proposition.

In other words, there isn't 'the world' about which there are discoveries. There are various worlds of understanding about which we seek to better comprehend.

[Daviddunn]

Unlike geometry, that formula is ultimately based on science; 

I have to disagree here. This formula proceeds from postulates too. The speed of light as the uttermost limit is postulated in the derivation of E=MC2. Scientific experiments have been conducted, and it was not possible for this speed to be exceeded. However, this does not prove that it cannot be exceeded.

Easy reference: emc2-explained.info/Emc2/Derive.htm#.U2 ... 2fmX32EbMI

[Londoner]

Daviddunn

You are quite right; they are also postulates although the postulates in physics could be verified or falsified empirically - sufficiently for the purposes of science - we might not ever be in a position to actually do so, but they are 'that sort of thing'.

But the postulates in geometry are just one set of alternatives. An alternative set would be equally valid, throwing up an alternative set of theorems, so there is no test that could show that one particular batch are the 'correct' ones.

[Daviddunn]

There are alternatives to euclidean geometry, and each has its field of application. Euclidean geometry is used in surveying, engineering and architecture. Another type of geometry is used in relativity. As to the test which can be applied for 'correctness' it will be usefullness, derived from a necessity to understand and operate in the world.  

In logic there are alternatives sets of assumptions at the foundational level, giving rise to different types of logical system.

Alternatives are not just for geometry. In physics also, there are competing alternatives. The well known is quantum mechanics and relativity, each has different sets of assumptions. Again, here too, usefulness is the criterion for 'correctness'.  

Postulates take the form of a universal judgement, I.e. everything or nothing. They cannot be verified empirically, neither in physics nor in geometry. At best empirical evidence give an indication that support the universal proposition of the postulate, and both physics and geometry have enough of these. At some point to take the path of knowledge, one must yield, and trust the evident. Any field of human inquiry has this in common, I.e. a point where one accepts the evident without proof. Not a single one is exempted, natural sciences, logic, mathematics (including geometry). Necessity is the law, and non contradiction is the guiding principle. If one denies this, the path of skepticism invites him/her to delusion and madness.

I must concede here, that Fafner's insistence got me into doing some serious thinking. I have been underestimating the power of reason when used to understand the world around.

[Londoner]

Daviddun

At some point to take the path of knowledge, one must yield, and trust the evident...If one denies this, the path of skepticism invites him/her to delusion and madness.

Yes, although that is to say something about us; what is necessary for us to operate.

That was Kant's point. Philosophy sought a 'synthetic a priori' that didn't require us to 'yield' or 'trust' - knowledge about the world that must be true. As you rightly point out, that is futile.

Kant argued that if we seek a 'synthetic a priori' it would better be understood as those necessary cognitive tools you mention, those that allow us to think at all.

[Daviddunn]

Yes, although that is to say something about us; what is necessary for us to operate. 

...to operate in the world. I believe this part is equally important. There are many things that can be said about Kant. Much has already been said. One thing for sure, he does not doubt the independent (of us) existence of the 'external world'. Ten years ago I read Kant, painful but worthy reading. He has had good points, some still stand but after the scientific discoveries that came after the Critique, the Critique itself stand in need of a critique.

Anyway, this exchange has been very enriching for me.