Philosophy Forums


7 + 5 = 12 as a synthetic statement

PrintPrint


Page: 1 2 3 4

7 + 5 = 12 as a synthetic statement
makerowner
Assistant Professor

Usergroup: Members
Joined: Apr 15, 2008

Total Topics: 11
Total Posts: 326
Posted 06/03/09 - 11:19 AM:
quote post
#21
keda wrote:
I think that explains it all. It requires a particular skill that isn't required when one simply analyzes a concept by looking at its subconstituents, namely, the skill of adding, multiplying, dividing and so on. We only come to 12 after adding 5 to 7, not by means of analyzing it.


Yes this is the key to the whole problem. Kant thought that we could distinguish analytic and synthetic judgments by simply considering the concepts involved; the fact that there is disagreement in this and other cases shows that he was wrong. A concept isn't like a box that either contains or doesn't contain other concepts; it's a word that can be used in certain ways and not (correctly) in others, and that's associated with other words in different ways.

For philosophy, Socrates, if pursued in moderation and at the proper age, is an elegant accomplishment, but too much philosophy is the ruin of human life. Even if a man has good parts, still, if he carries philosophy into later life, he is necessarily ignorant of all those things which a gentleman and a person of honour ought to know.
keda
Ijon Tichy
Avatar

Usergroup: Members
Joined: Jul 25, 2005
Location: Finland

Total Topics: 38
Total Posts: 3640
Posted 06/03/09 - 02:49 PM:
quote post
#22
makerowner wrote:

Kant thought that we could distinguish analytic and synthetic judgments by simply considering the concepts involved; the fact that there is disagreement in this and other cases shows that he was wrong.

I don't see how that follows.

All about making money
Free Europe Now How to fix your country
In thought, men distance themselves from nature in order thus imaginatively to present it to themselves--but only in order to determine how it is to be dominated - Adorno and Horkheimer
makerowner
Assistant Professor

Usergroup: Members
Joined: Apr 15, 2008

Total Topics: 11
Total Posts: 326
Posted 06/04/09 - 09:22 AM:
quote post
#23
keda wrote:

I don't see how that follows.


If determining whether a judgement was analytic or synthetic was a simple matter of examining our concepts through introspection, then it should be intuitively obvious for any given judgement which one it is. Which isn't to say that there couldn't be some cranks who disagreed (just as there are people who say things like "time is an illusion", etc.); but they would be recognizable as cranks, and the vast majority of us would agree (just as anyone who accepts the validity of the distinction will recognize "all triangles have three sides" as analytic).

For philosophy, Socrates, if pursued in moderation and at the proper age, is an elegant accomplishment, but too much philosophy is the ruin of human life. Even if a man has good parts, still, if he carries philosophy into later life, he is necessarily ignorant of all those things which a gentleman and a person of honour ought to know.
Aaron T Bentley
Initiate

Usergroup: Members
Joined: Dec 04, 2006

Total Topics: 0
Total Posts: 1
Posted 06/04/09 - 09:47 AM:
quote post
#24
There is a simple solution to this problem which I have not yet seen brought up. In terms of sense and reference, this is a synthetic judgement because the thought is a synthetic thought. One must have a conceptual understanding of addition and a understand the objects 7 and 5 to understand that when combined into a single collaborative thought, they refer to 12.
YadaYada
fool
Avatar

Usergroup: Members
Joined: Jun 12, 2004
Location: in the past

Total Topics: 5
Total Posts: 316
Posted 06/04/09 - 05:47 PM:
quote post
#25
I would think that two and four are familiar proper names, and that two plus two is four is just recalled from having memorized the problem. This is just as analytic as bachelors are.

With more maths it gets more complicated.
Applying the rules of math as learned in grade school is straight-forward analytic shorthand for 1+1+...
Proving anything is synthetic because possibly novel combinations of logical rules must be introduced.
Inventing new math is synthetic since creative introduction of axioms is necessary here.

{}{}{}{ . . . }{}{}{ . . . }{}{}{}{}{ . . .
We may assume the superiority ceteris paribus of the demonstration which derives from fewer postulates ~ Aristotle
keda
Ijon Tichy
Avatar

Usergroup: Members
Joined: Jul 25, 2005
Location: Finland

Total Topics: 38
Total Posts: 3640
Posted 06/05/09 - 02:25 AM:
quote post
#26

makerowner wrote:
keda wrote:

I don't see how that follows.


If determining whether a judgement was analytic or synthetic was a simple matter of examining our concepts through introspection, then it should be intuitively obvious for any given judgement which one it is. Which isn't to say that there couldn't be some cranks who disagreed (just as there are people who say things like "time is an illusion", etc.); but they would be recognizable as cranks, and the vast majority of us would agree (just as anyone who accepts the validity of the distinction will recognize "all triangles have three sides" as analytic).


I'm still not sure what your argument is. If you're saying that Kant said that most people would be recognizable as cranks and that the vast majority of us would agree then I think you may be misrepresenting him. We're not voting on what is true after all.

All about making money
Free Europe Now How to fix your country
In thought, men distance themselves from nature in order thus imaginatively to present it to themselves--but only in order to determine how it is to be dominated - Adorno and Horkheimer
makerowner
Assistant Professor

Usergroup: Members
Joined: Apr 15, 2008

Total Topics: 11
Total Posts: 326
Posted 06/06/09 - 05:07 PM:
quote post
#27



keda wrote:






makerowner wrote:
If determining whether a judgement was analytic or synthetic was a simple matter of examining our concepts through introspection, then it should be intuitively obvious for any given judgement which one it is. Which isn't to say that there couldn't be some cranks who disagreed (just as there are people who say things like "time is an illusion", etc.); but they would be recognizable as cranks, and the vast majority of us would agree (just as anyone who accepts the validity of the distinction will recognize "all triangles have three sides" as analytic).




I'm still not sure what your argument is. If you're saying that Kant said that most people would be recognizable as cranks and that the vast majority of us would agree then I think you may be misrepresenting him. We're not voting on what is true after all.


No that's not at all what I meant. I'll try to rephrase it. Kant thought that we could distinguish analytic judgements from synthetic ones just by mentally examining the concepts involved. My claim is that if that were true, we would expect there to be near-unanimity about which kind a particular judgement is, just as there is unanimity about "all bachelors are unmarried" being analytic (among those who accept the distinction between analytic and synthetic). Mathematical equations have been claimed as both synthetic and analytic by significant groups (ie. not just isolated cranks, like people who claim that time is an illusion), which makes Kant's claim at least questionable.


For philosophy, Socrates, if pursued in moderation and at the proper age, is an elegant accomplishment, but too much philosophy is the ruin of human life. Even if a man has good parts, still, if he carries philosophy into later life, he is necessarily ignorant of all those things which a gentleman and a person of honour ought to know.
keda
Ijon Tichy
Avatar

Usergroup: Members
Joined: Jul 25, 2005
Location: Finland

Total Topics: 38
Total Posts: 3640
Posted 06/07/09 - 01:09 AM:
quote post
#28
makerowner wrote:
No that's not at all what I meant. I'll try to rephrase it. Kant thought that we could distinguish analytic judgements from synthetic ones just by mentally examining the concepts involved. My claim is that if that were true, we would expect there to be near-unanimity about which kind a particular judgement is, just as there is unanimity about "all bachelors are unmarried" being analytic (among those who accept the distinction between analytic and synthetic). Mathematical equations have been claimed as both synthetic and analytic by significant groups (ie. not just isolated cranks, like people who claim that time is an illusion), which makes Kant's claim at least questionable.

I don't think it does, and your argument seems to be analoguous to one in favour of nihilism, where it the case that scientists happen to disagree, which they in fact do. I suggest mathematical equations are either interpreted differently by various groups, and so they form different judgmenents i.e. their usage of the symbols differ, or they use them in the same manner yet one group is just plain wrong regarding its analyticity. Given that none of the groups who claim it is analytical has been able to put forth definitions of the symbols, whereby they may prove its analyticity, I suggest that they are not even plain wrong, but unqualified to make a judgment.

All about making money
Free Europe Now How to fix your country
In thought, men distance themselves from nature in order thus imaginatively to present it to themselves--but only in order to determine how it is to be dominated - Adorno and Horkheimer
makerowner
Assistant Professor

Usergroup: Members
Joined: Apr 15, 2008

Total Topics: 11
Total Posts: 326
Posted 06/07/09 - 09:08 AM:
quote post
#29

keda wrote:
I don't think it does, and your argument seems to be analoguous to one in favour of nihilism, where it the case that scientists happen to disagree, which they in fact do. I suggest mathematical equations are either interpreted differently by various groups, and so they form different judgmenents i.e. their usage of the symbols differ, or they use them in the same manner yet one group is just plain wrong regarding its analyticity. Given that none of the groups who claim it is analytical has been able to put forth definitions of the symbols, whereby they may prove its analyticity, I suggest that they are not even plain wrong, but unqualified to make a judgment.


I'm not sure quite what you mean by 'nihilism'. (It seems to be able to mean anything that the person writing doesn't like.) But in any case the argument I gave above doesn't apply to natural science because natural science doesn't claim intuitive knowledge of its objects. Since natural science studies non-mental objects, scientists' disagreeing is just an indication that there's more to be learned about the object in question. Kant's doctrine on analytic/synthetic judgments, on the other hand, says that we only have to mentally examine our concepts in order distinguish one kind of judgement from the other; ie. it's not a question of collecting more data, as it is in natural science.


If I say that mathematical equations are analytic, and you say they're synthetic (I'm just using this for exposition; I'm not taking one side or the other), and the source of my knowledge is introspection, then how can I explain your disagreement? I would have to a) claim that you're deliberately misrepresenting the results of your introspection; or b) claim that you are somehow less able to introspect than I am; or c) claim that you don't have the same concepts as I do. Obviously a) and b) are pretty weak responses, since you could say the exact same thing to me and we would be at an impass. One of your suggestions ("I suggest mathematical equations are [...] interpreted differently by various groups, and so they form different judgmenents i.e. their usage of the symbols differ") seems to be something like c); the problem with this proposal is that the only reason to believe that our concepts differ is our disagreement. Our concepts of numbers agree when we're just using them (ie. to perform mathematical operations) rather than talking about them, so the only difference between my concepts and yours is that I think mine make analytic judgments and you think yours make synthetic judgements. (Again, I'm just using this as an example; I'm not taking sides.) Describing a judgement as 'analytic' or 'synthetic' would then say nothing about it, since it would have exactly the same use either way. Ascribing the disagreement to our having different concepts just makes the concepts meaningless.


Your second suggestion ("I suggest that they [the groups who claim it is analytical] are not even plain wrong, but unqualified to make a judgment") looks like my b) above. The question here is how is it that you have such a superior analyticity-detecting ability? How can I know that you're the one who has it, and not them? And as for the "definitions of the symbols", doesn't defining numbers as "1+1+1+..." make mathematical equations analytic? If 2 is defined as 1+1, and 4 is defined as 1+1+1+1, then 2+2=4 would just be a tautology, 1+1+1+1=1+1+1+1. (Philosophy of math isn't really a big interest for me, so though I'm sure someone has come up with an objection to this already, I don't know it.)


If, instead of assuming analyticity can be determined by introspection, we look at concepts as normative standards for the use of words, then this disagreement isn't problematic. It could either be that we don't know enough about the norms in question (just as natural scientists disagree about objects they don't know enough about yet); or it could be that the norm doesn't cover this particular case, and then it would be philosophy's job to establish a norm, if needed (just as the courts in a common-law system don't just determine whether this particular case falls under the concept of, eg. 'theft', but also change the concept of 'theft' by deciding what cases fall under it).


For philosophy, Socrates, if pursued in moderation and at the proper age, is an elegant accomplishment, but too much philosophy is the ruin of human life. Even if a man has good parts, still, if he carries philosophy into later life, he is necessarily ignorant of all those things which a gentleman and a person of honour ought to know.
keda
Ijon Tichy
Avatar

Usergroup: Members
Joined: Jul 25, 2005
Location: Finland

Total Topics: 38
Total Posts: 3640
Posted 06/08/09 - 01:35 AM:
quote post
#30
makerowner wrote:

I'm not sure quite what you mean by 'nihilism'. (It seems to be able to mean anything that the person writing doesn't like.)

Nihilism is an extreme form of scepticism that denies all basis for objective truth.

But in any case the argument I gave above doesn't apply to natural science because natural science doesn't claim intuitive knowledge of its objects.

I'm making an analogy with natural science, not suggesting natural science is an application. Just because scientists disagree doesn't mean that there is no basis for science, in the same manner just because people disagree about the analyticity and syntheticity of judgements, doesn't mean there is no basis for this distinction. A lot of people can be wrong about something. Its nothing unusual.

One of your suggestions ("I suggest mathematical equations are [...] interpreted differently by various groups, and so they form different judgmenents i.e. their usage of the symbols differ") seems to be something like c); the problem with this proposal is that the only reason to believe that our concepts differ is our disagreement. Our concepts of numbers agree when we're just using them (ie. to perform mathematical operations) rather than talking about them, so the only difference between my concepts and yours is that I think mine make analytic judgments and you think yours make synthetic judgements. (Again, I'm just using this as an example; I'm not taking sides.) Describing a judgement as 'analytic' or 'synthetic' would then say nothing about it, since it would have exactly the same use either way.

The difference of analytic and synthetic judgments *is* a difference in use, since analyticity of a judgment is the quality of having its validity depend on its concepts alone, while the syntheticity of a judgement is the quality of having its validity depend on something external to its concepts. In other words, when you make an analytic judgment, you do so by merely analyzing its concepts, while when you make a synthetic judgment, you do so by appealing to something external to its concepts. This is different usage. If our usage does not differ, our judgments are either both synthetic, or both analytic, and one of us must be wrong.

Your second suggestion ("I suggest that they [the groups who claim it is analytical] are not even plain wrong, but unqualified to make a judgment") looks like my b) above. The question here is how is it that you have such a superior analyticity-detecting ability?

It is merely the inability here to form a certain judgment in the first place. If you have no concept of what you are talking about, you are basically speaking gibberish in your own eyes and not trying to communicate anything. It is not so farfetched as some do treat math as some kind of mechanical process that "just works". When they solve equations, they have no idea what they mean until the final result is given. It doesn't have to mean they aren't good at seeing how their judgements are analytic, once they have actually made them, but just writing something down or making a sound doesn't count as making a judgment. To do so you must have a concept of what the symbols actually mean.

And as for the "definitions of the symbols", doesn't defining numbers as "1+1+1+..." make mathematical equations analytic? If 2 is defined as 1+1, and 4 is defined as 1+1+1+1, then 2+2=4 would just be a tautology, 1+1+1+1=1+1+1+1. (Philosophy of math isn't really a big interest for me, so though I'm sure someone has come up with an objection to this already, I don't know it.)

This or something similar has been suggested before, and I have suggested this is a play on ambiguety, since the former "1+1+1+1" look visually the same as the latter "1+1+1+1" but they may be semantically different if 4 isn't actually defined as (1+1)+(1+1). But if this is the case, you cannot do the same with 1+3=4. Remember, visual aid must not be necessary to validate an analytic judgment.


If, instead of assuming analyticity can be determined by introspection, we look at concepts as normative standards for the use of words, then this disagreement isn't problematic.

Kant defined concepts as cognitions, so I think it is a misrepresentation to say he was wrong, simply because their analyticity cannot be determined by introspection. You object and say that introspection is not sufficient to solve disgreements, however if I need to determine the analyticity of my judgments, I would rather not have to ask someone else about it each time. Different horses for different courses. If we need to solve a disagreement, we invent conventions and try to make clear our usage. If someone intends to not to do so, but deliberately uses confusing and ambiguous language in order to manipulate others, it isn't a particularly unique problem for me.

All about making money
Free Europe Now How to fix your country
In thought, men distance themselves from nature in order thus imaginatively to present it to themselves--but only in order to determine how it is to be dominated - Adorno and Horkheimer
Download thread as

Page: 1 2 3 4



Sorry, you don't have permission to post. Log in, or register if you haven't yet.