What Justifies an A Priori Concept Like Infinity?

[Willowz]

What justifies an a priori statement? If not experience, then what else is left?
I mean, I get the example with counting numbers by adding another number to that last number, thus we arrive at a concept called "infinity". How is that an a priori example?

I'm not getting the big picture here.

 

[Willowz]

So, nobody knows what an a priori is? Or maybe I don't know... OK let's see:

Varieties of Judgment

In the Prolegomena to any Future Metaphysic (1783) Kant presented the central themes of the first Critique in a somewhat different manner, starting from instances in which we do appear to have achieved knowledge and asking under what conditions each case becomes possible. So he began by carefully drawing a pair of crucial distinctions among the judgments we do actually make.

The first distinction separates a priori from a posteriori judgments by reference to the origin of our knowledge of them. A priori judgments are based upon reason alone, independently of all sensory experience, and therefore apply with strict universality. A posteriori judgments, on the other hand, must be grounded upon experience and are consequently limited and uncertain in their application to specific cases. Thus, this distinction also marks the difference traditionally noted in logic between necessary and contingent truths.

 

[GeorginaS]

Well, okay, understanding what you're idea of a priori knowledge is would help. If you're talking in terms of mathematics, then a priori knowledge tends to mean knowledge that is gained by deduction rather than by empirical evidence.

So, for example, with math, you can think about having two pencils in front of you now and, when Bob brings you three more pencils, you can deduce that you'll then have five. You've arrived at that by way of deduction and don't actually have to place two pencils in front of you and then add another three to know how many you'll have then.

The other use for a priori, though, is "innate" or "inborn" knowledge. There is an argument that I think is rather flawed that asserts that mathematics is true without reference to reality. That the knowledge of mathematics -- as opposed to the knowledge created by mathematics -- is known without reference to reality.

 

[Willowz]

Well, okay, understanding what you're idea of a priori knowledge is would help. If you're talking in terms of mathematics, then a priori knowledge tends to mean knowledge that is gained by deduction rather than by empirical evidence.

Deduction of what, if not empirical evidence?

So, for example, with math, you can think about having two pencils in front of you now and, when Bob brings you three more pencils, you can deduce that you'll then have five. You've arrived at that by way of deduction and don't actually have to place two pencils in front of you and then add another three to know how many you'll have then.

I will not know if he brought me in total 5 pencils until I see them all laid down together.

The other use for a priori, though, is "innate" or "inborn" knowledge. There is an argument that I think is rather flawed that asserts that mathematics is true without reference to reality. That the knowledge of mathematics -- as opposed to the knowledge created by mathematics -- is known without reference to reality.

I have a feeling that a priori is in other words our capacity rather than an ambiguous terms such as innate or inborn. But h o w do we measure this capacity?

 

[Goethe]

Deduction of what, if not empirical evidence?

Of definition. If I play an A minor triad on a piano, you know that the notes I played were A-C-E. You don't have to walk over and look at my fingers because by definition an A minor triad consists of A-C-E.

I will not know if he brought me in total 5 pencils until I see them all laid down together.

Granted. But we're going on the assumption that you have two and he's bringing you three. Can you conceive of a scenario where you have two pencils, and someone brings you three, and yet the pencils he brings and the ones you have somehow add up to six?

 

[lompocus]

I wouldn't consider either of those cases a priori things. Each is, really, deduced by empirical evidence: You're taking an event (data input) and expecting a pattern that has emerged to continue. You just happen to have a very limited data set... Or, I have the idea of an a priori statement mixed up with something else.

Remove the the association of an 'innate "truth" ' in the definition of the thing and what you have left is a statement that is taken to be true not by the gathering of any evidence whatsoever. Mathematical proofs come to mind. You can derive something from them that is fully functional in a mathematical construct that adheres to certain rules, else, the reasoning (the proof) fails. It just so happens that the math could also be derived from a bunch of data (or not, but the line blurs when you get to applications of any kind of math, i.e. physics...).

It seems silly to have two strict definitions of what is a priori and what isn't, though. They inherently complement each other (i.e. very simple kinematics).

 

[Goethe]

Now for me a priori is more or less synonymous with 'by definition', because that just seems like a more useful way to apply the phrase. But this caught my interest:

Each is, really, deduced by empirical evidence: You're taking an event (data input) and expecting a pattern that has emerged to continue.

My understanding of empiricism is that it refers solely to things which we observe through our senses. But you've related it to 'data-input'.

So let's take my perceptions. I know (i.e. have stored data) that I am perceiving things right now - that is, I don't just know what I see, but I know that I see. Do I perceive that I perceive? Or, do I 'just know' that I have perceptions? Or something else?

I don't presume to have the answer to this question, btw

 

[lompocus]

I doubt the question doesn't have an answer, but that it's just too complicated for us to fully comprehend at the moment.

This unknown, of course, becomes a bit clearer if one of the amazing scientists around here would kindly explain how in the world our brain is functioning!

Going back to Willow's quote (Kemp's definition), the data that I would consider for something related to what seems to be under this very broad and vague 'a priori' thing (a fault of most non-modern, and a lot of modern, "philosophy") is mostly separated from human senses. You bring up that we, ultimately, need to interpret some group of data in some way. Therefore, why is something commonsense disregarded?

I think that the idea is not to remove the human factor, but to remove those commonsense, and usually terribly wrong, ideas from the (a priori) statement. I think, by saying 'empirical evidence' I made things a little confusing. What I meant was to only consent to using data in formulating some (temporary) conclusion so long as it can be quantified. Can you mathematically represent it? Unfortunately I feel that this explanation is also a bit vague, but hopefully someone comes along with a better idea.

This whole a priori business really starts to become important when put into the context of the decisions made by people in power (politicians). The senate would have been more prompt to deal with the healthcare issue (and would have been more comprehensive in dealing with it, too!), if its members understood the information brought to them. Sadly, what we do end up getting is a bunch of unfounded judgments and plenty of unnecessary and uncompromising partisanship. But this is only one little example.

 

[heusdens]

Can you conceive of a scenario where you have two pencils, and someone brings you three, and yet the pencils he brings and the ones you have somehow add up to six?

Such a scenario is very well plausible, when applied for instance to counting clouds. They can easily merge together in which the sum total of clouds is not equal to the individual count of clouds.

 

[heusdens]

Counting rabbits or mice -or for that purpose: bacteria - is another example. They don't add up, they multiplicate!

 

[disregardthat]

An a priori statement is not a statement about the world, it is an expansion of an already known concept. We don't need data to back it up, because the statement doesn't refer to something which makes sense to back up empirically.

One classical example of an a priori conclusion is that all bachelors are unmarried. This does not refer to the bachelors themselves, but rather to the state of being a bachelor. The state of being a bachelor is exactly that of being an unmarried man. So the a priori conclusion is a tautology, it simply cannot be false. We need no evidence to back that up.

When we are counting clouds the a priori conclusion we arrive at depends on the logical picture we have of the situation. We quantify clouds in our mind, and the logical conclusions only hold to the degree clouds can be quantified. The conclusions based on a physical model are always a priori, but the question of whether the conclusions correspond to reality is the same question as whether the model actually represents reality.

So we can say that all statements about the world are a posteriori, while a statement deduced from a logical picture is a priori, but the logical picture of a situation does not necessarily correspond to the situation itself.

 

[disregardthat]

The other use for a priori, though, is "innate" or "inborn" knowledge. There is an argument that I think is rather flawed that asserts that mathematics is true without reference to reality. That the knowledge of mathematics -- as opposed to the knowledge created by mathematics -- is known without reference to reality.

If I understand you correctly:
Mathematics is true without reference to reality! All mathematical conclusions are arrived at through logical proof, and all mathematical inferences therein are in reference to the given axioms, not reality.

If the axioms can be used in a given physical model or not is not relevant to mathematics itself, but to the use of mathematics in physics. What we deduce from the axioms does not depend on the physical situation they may represent.

 

[Willowz]

An a priori statement is not a statement about the world, it is an expansion of an already known concept.

So, concepts are unrelated to the world? Does this lead us to some Platonism?

So we can say that all statements about the world are a posteriori, while a statement deduced from a logical picture is a priori, but the logical picture of a situation does not necessarily correspond to the situation itself.

"...the logical picture of a situation does not necessarily correspond to the situation itself" How can you tell?

 

[Pythagorean]

Of definition. If I play an A minor triad on a piano, you know that the notes I played were A-C-E. You don't have to walk over and look at my fingers because by definition an A minor triad consists of A-C-E.



Granted. But we're going on the assumption that you have two and he's bringing you three. Can you conceive of a scenario where you have two pencils, and someone brings you three, and yet the pencils he brings and the ones you have somehow add up to six?

They're mechanical pencils, and one has twice the lead, and all you care about is how much lead you have to write with.

Maybe you were thinking of wooden pencils and the textual definition of pencil as an object, not based on logistics.

Either way, the point is that you have to qualify it. There would be no way to ever conceive of math if we couldn't empirically experience a world where matter is conserved. 2 pencils + 3 pencil = 5 pencils would be as meaningful as 2 pencils + 8 pencils = -3 pencils.

My conclusion: purely quantitative descriptions are meaningless, while purely qualitative descriptions are ambiguous. There's a harmony between the two.

 

[disregardthat]

So, concepts are unrelated to the world? Does this lead us to some Platonism?

We think like this all the time. What happens when you are imagining a move in the game of chess? You are drawing necessary conclusions based on the rules of the game. You don't need the actual game to do this, but does that mean the game you are thinking of must be located in a platonic realm? No, it does not; this has nothing to do with platonism.

"...the logical picture of a situation does not necessarily correspond to the situation itself" How can you tell?

Experience shows us if our way of thinking of a situation is reasonable or not. We can easily count stones, but we are always careful when we are counting drops of water.

 

[Goethe]

They're mechanical pencils, and one has twice the lead, and all you care about is how much lead you have to write with.

Maybe you were thinking of wooden pencils and the textual definition of pencil as an object, not based on logistics.

Precisely. The whole point is that once you have a working definition of something, you can infer things based on that definition. The first paragraph doesn't really disrupt this, because in that case 'pencil' means 'a certain amount of graphite'. Same with the example with clouds earlier in this thread - you either define 'cloud' as a certain amount of vapor, or you just ask how many there are before they merge.

A lot of us are talking past each other in this thread. I'll say that I don't deny that all of our knowledge has an empirical origin in principle. But as soon as you start using language, you open the door to analytic statements. And you can go quite a ways just on definition, as mathematics shows.

 

[Pythagorean]

Precisely. The whole point is that once you have a working definition of something, you can infer things based on that definition. The first paragraph doesn't really disrupt this, because in that case 'pencil' means 'a certain amount of graphite'. Same with the example with clouds earlier in this thread - you either define 'cloud' as a certain amount of vapor, or you just ask how many there are before they merge.

So then you're stance is that your definition is based on how it's written in a dictionary, based on the text, not the meaning the author was trying to convey with it? And that the textual definition is the 'right' definition? (see: http://en.wikipedia.org/wiki/Textualism" )

Cloud:
What if two clouds ARE merging when you freeze the frame? At one percentage of overlap do you consider them one cloud? Do you think your discrepancy is universal?

 

[disregardthat]

A large pencil can contain twice the amount of graphite, but we still consider it one pencil.

In language are definitions fleeting, and only precise if the situation requires it. Definitions are commonly a way of describing how a word is used, but the usage does not flow from the definitions. It's the other way around.

We don't need a definition of clouds to know what a cloud is, but you will need to know what a cloud is to define it.

 

[Goethe]

So then you're stance is that your definition is based on how it's written in a dictionary, based on the text, not the meaning the author was trying to convey with it?

I mean whatever definition we agree on. I defaulted to the textual example of 'pencil' because the original context didn't seem to imply anything else.

I don't think I clearly understand you. Are you saying that things inferred from definition ('a priori') won't be clear in a discussion between two people, or that we can't make real-world inferences about things because they may not fit our definition? Or am I missing the boat entirely here?

 

[Willowz]

We think like this all the time. What happens when you are imagining a move in the game of chess? You are drawing necessary conclusions based on the rules of the game. You don't need the actual game to do this, but does that mean the game you are thinking of must be located in a platonic realm? No, it does not; this has nothing to do with platonism.

How could I play a game while not playing by the rules?

 

[Willowz]

About definitions.
Where are we pulling out all these definitions from?
How can it be, that all bachelors are unmarried?

 

[Goethe]

About definitions.
Where are we pulling out all these definitions from?

see below :)

 

[Pythagorean]

I mean whatever definition we agree on. I defaulted to the textual example of 'pencil' because the original context didn't seem to imply anything else.

I don't think I clearly understand you. Are you saying that things inferred from definition ('a priori') won't be clear in a discussion between two people, or that we can't make real-world inferences about things because they may not fit our definition? Or am I missing the boat entirely here?

The point is that mathematics comes from empirical observations, as willows is trying to convey. You never responded to this part of my post, which is where I cleary stated my point. Not sure if that meant you agreed with it, didn't see it, or thought another point you made hinged on it, etc, etc. (a lack of apriori :P).

Either way, the point is that you have to qualify it. There would be no way to ever conceive of math if we couldn't empirically experience a world where matter is conserved. 2 pencils + 3 pencil = 5 pencils would be as meaningful as 2 pencils + 8 pencils = -3 pencils.

My conclusion: purely quantitative descriptions are meaningless, while purely qualitative descriptions are ambiguous. There's a harmony between the two.

 

[Goethe]

I knew something was wrong here. Earlier on I said,

I don't deny that all of our knowledge has an empirical origin

Of course it eventually goes back to observation, but sometimes (as in mathematics) we stay in 'mental space' so long we seem to be dealing with a different kind of knowledge. My argument was essentially that 'a priori' is a useful/appropriate designation for this knowledge.

 

[Pythagorean]

I knew something was wrong here. Earlier on I said,


Of course it eventually goes back to observation, but sometimes (as in mathematics) we stay in 'mental space' so long we seem to be dealing with a different kind of knowledge. I do this to give the phrase 'a priori' a useful... Definition.

Ok, I don't disagree. I think of mathematics as a kind of logic structure that we can build models with. I'm bias of course, I've never taken a proofs class; I only take classes necessary for sciences.

 

[Willowz]

see below :)

Surely, some bachelors are married...

 

[Willowz]

The point is that mathematics comes from empirical observations, as willows is trying to convey.

It would be a comforting thought to think so, but the half-*** example I provided in the OP(Infinity as a limiting concept) baffles me. That we can know our own limits without empirical evidence!
How do you explain such a thing??

 

[disregardthat]

How could I play a game while not playing by the rules?

What does that have to do with what you quoted?

And surely, no bachelors could be married, for that would strip them of their status as bachelors. Don't you see what's going on here?

 

[yossell]

The definition of a priori is normally in terms of knowledge or justification:

Roughly, an a priori truth is one which *could* be known without recourse to experience, while an a posteriori truth is one which requires some kind of experience for someone to know it.

Caveats are needed because some experience may be required to gain the relevant concepts. For instance, we may need experience of green things to have the concept `green'. But, once we have the concept, we can know that everything is green or not green without having to go out and check or do experiments.

So it may be true that experience is required for us to form the relevant concepts - say the concepts of a number of things, but that doesn't alone imply that mathematical truths are not a priori any more than the fact we need experience of green things means 'everything is green or not green' is not a priori.

Note that an a priori truth could be known a posteriori. Somebody may find out that everything is green or not green by going out and checking; somebody may come to know that 2 and 2 is always 4 by counting a few cases, and then generalising by induction.

 

[Willowz]

What does that have to do with what you quoted?

Maybe I should ask you... What does this mean in your previous post?

You are drawing necessary conclusions based on the rules of the game. You don't need the actual game to do this...

 

[Willowz]

And surely, no bachelors could be married, for that would strip them of their status as bachelors. Don't you see what's going on here?

What I don't get with the bachelors example is the ideal situation that is presented to us. How can it be that all bachelors are unmarried? SO many things in reality elude our faculties of reasoning but bachelors being unmarried is almost a universal truth to us, how is this so?

 

[Willowz]

So, another question. Can the so called "a priori" truths be only made from within a complete system(Godel)? In this case with the bachelors? It seems so. Should I start a new thread regarding this question?

 

[disregardthat]

What I don't get with the bachelors example is the ideal situation that is presented to us. How can it be that all bachelors are unmarried? SO many things in reality elude our faculties of reasoning but bachelors being unmarried is almost a universal truth to us, how is this so?

A bachelor is defined as being an unmarried male. Therefore, if we are going to call anyone a bachelor, he necessarily have to be unmarried, or else we would have contradicted our own definition.

Suppose for a moment that a person is a bachelor and married. That implies he is married and not married at the same time. Is this still not a convincing argument for that no bachelor can be married?

Maybe I should ask you... What does this mean in your previous post?

You are drawing necessary conclusions based on the rules of the game. You don't need the actual game to do this...

By knowing the rules of a game, you can imagine possible moves. These moves are necessarily possible given the rules. Hence you are drawing necessary conclusions. These a priori conclusions are not subject to empirical testing, they are contained in the premises (rules), i.e. inherent to the rules which you assume; in the same way as being unmarried is contained within being a bachelor.

 

[Willowz]

A bachelor is defined as being an unmarried male. Therefore, if we are going to call anyone a bachelor, he necessarily have to be unmarried, or else we would have contradicted our own definition.

On what basis can we make this definition? How is it, that this definition applies so strictly to the situation given.

By knowing the rules of a game, you can imagine possible moves. These moves are necessarily possible given the rules. Hence you are drawing necessary conclusions. These a priori conclusions are not subject to empirical testing, they are contained in the premises (rules), i.e. inherent to the rules which you assume; in the same way as being unmarried is contained within being a bachelor.

This is all saying that I am all the time playing the game(following the rules). But, what you said earlier gave me the impression that I wasn't playing the game even if I were playing by the rules: "You are drawing necessary conclusions based on the rules of the game. You don't need the actual game to do this..."

 

[disregardthat]

On what basis can we make this definition? How is it, that this definition applies so strictly to the situation given.

We can define whatever we want, and no situation was given. By the a priori conclusion we are expanding (or exploring) our notion of the term 'bachelor'. It does not have to apply to any situation. No bachelor would be married even if there was not such thing as unmarried males, or males at all. It is purely a semantical argument without reference to reality.

This is all saying that I am all the time playing the game(following the rules). But, what you said earlier gave me the impression that I wasn't playing the game even if I were playing by the rules: "You are drawing necessary conclusions based on the rules of the game. You don't need the actual game to do this..."

You can imagine possible moves in chess without having the physical game in front of you. That ought be obvious from the context.