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7 + 5 = 12 as a synthetic statement

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7 + 5 = 12 as a synthetic statement
keda
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Posted 05/27/09 - 02:38 PM:
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xzJoel wrote:

It seems to me that "12 is not the addition of 7 and 5" is a contradiction because by equivalence "12 is not the addition of 7 and 5" is the same as "the addition of 7 and 5 is not the addition of 7 and 5".

by equivalence 12 is the addition of 7 and 5? Aren't you begging the question that this is an analytic judgment?
It seems to me that "tables are not yellow" is a contradiction, because by equivalence "tables are not yellow", is the same as "yellow things are not yellow"
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xzJoel
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Posted 05/27/09 - 02:51 PM:
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keda wrote:

by equivalence 12 is the addition of 7 and 5? Aren't you begging the question that this is an analytic judgment?
It seems to me that "tables are not yellow" is a contradiction, because by equivalence "tables are not yellow", is the same as "yellow things are not yellow"


I am not quite certain what you mean.

I intended to convey that using the rules of replacement and equivalence, anytime we are using number we can replace a number with an operation.

x is y. Where "x" is a subject and "y" is a definition.
x = a&b
(a&b) is y.

Do you disagree that a definitional statement of "x is y" is analytic?
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keda
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Posted 05/27/09 - 03:23 PM:
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There are two ways x can be y, either x is y because y is contained in x or it isn't. To say 12 is 7+5 is analytic because 12 is defined as 7+5 is to beg the question as to whether it is contined within the subject.
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Yahadreas
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Posted 05/27/09 - 03:32 PM:
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keda wrote:
To say 12 is 7+5 is analytic because 12 is defined as 7+5 is to beg the question as to whether it is contined within the subject.


Then it would also beg the question to say that the claim "all bachelors are unmarried men" is analytic because a bachelor is defined as an unmarried man.
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xzJoel
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Posted 05/27/09 - 03:41 PM:
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keda wrote:
There are two ways x can be y, either x is y because y is contained in x or it isn't. To say 12 is 7+5 is analytic because 12 is defined as 7+5 is to beg the question as to whether it is contined within the subject.


My hope was to suggest that the concept of "12" could be conceptualized in different ways, thereby changing the analytical status of "7+5=12" depending on which concept you chose to adopt. If I failed in this, I apologize.

More words:

Mere semantic representation of a concept (or symbolic representation such as "2" vs. "II") does not change a concept. You must come to understand a concept as being something in particular. If you have a concept which has an equivalent concept, that does not make them the same concept, nor does joining them in a sentence make it an analytic statement. If a concept contains (perhaps "entails"?) another concept, joining them together may permit an analytic statement. However, the way you conceive of a concept may be dynamic such that you have multiple ways of conceiving of the same concept.

A statement that is analytic so long as you maintain one conception of the subject of the analytic statement may very well become synthetic if you change your conception of the subject. There is nothing inherently correct about choosing one "basic" concept as your starting point rather than another. (This "basic concept" is what is referred to as primaries, no?)

Do I make more sense now?


If you believe me to be incorrect in my understanding of analytic vs. synthetic and the relationship to conceptually equivalent terms vs. multiple ways of conceiving of the same concept, please feel free to correct me or say, "You are so wrong I do not know where to begin. Stop spreading misinformation." I claim no expertise.
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Yahadreas
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Posted 05/27/09 - 03:47 PM:
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An interesting idea:

Let's accept that the concept "12" contains "7 + 5". Just for the sake of argument. This is so because we have done some addition and said that "instead of writing 1 + 1 + 1... + 1, I am going to write 12". The definition of 12, then, is 7 + 5 (with these two being defined by their associated addition of ones).

However, I'm hesitant to say it's the case that 3 * 4 is contained in 12. 12 is defined by adding ones. Not by multiplication. The multiplication is perhaps such a synthetic a priori.
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worldlogicleague
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Posted 05/27/09 - 04:04 PM:
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Yahadreas wrote:


Then it would also beg the question to say that the claim "all bachelors are unmarried men" is analytic because a bachelor is defined as an unmarried man.


So in the same way would you say "the claim 7+5=12 is analytic because the subject concept "7+5" is defined as a 12"

If this were so I'd agree, but is the concept "7+5" defined as 12? If so 12 would have to be literally found necessarily in the definition of 7+5, in the same way that:

Def. bachelor -- an unmarried male.

Prop. All Bachelors are unmarried males.

Since the predicate is contained in the subject it's an analytical statement.

But i gave a sufficient definition of 7+5 as an additive operation where 7 is the first number and 5 is the second.

And there is no mention of nor a need for 12 in the subject definition.

Or imagine an experiment where we go to the jungles of Amazon and find a 5 year old boy with no math training. We teach him the concept 7+5 using my definition. So he knows the concept: its some sort of adding operation using the numbers 7 and 5. But he doesn't know any math skills. By just knowing the concept 7+5, will he immediately, suddenly blurt out 12, or 7+5 = 12 because it was contained in the concept we taught him? I don't think he will.

Now imagine we teach him the concept bachelor -- an unmarried male. With just this concept he could suddenly blurt out All Bachelors are unmarried males, because it was contained in the subject we taught him - as part of its definition.

Edited by worldlogicleague on 05/27/09 - 04:11 PM
xzJoel
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Posted 05/27/09 - 04:14 PM:
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worldlogicleague wrote:


Now imagine we teach him the concept bachelor -- an unmarried male. With just this concept he could suddenly blurt out All Bachelors are unmarried males, because it was contained in the subject we taught him - as its definition.


Are you saying that what makes something "analytic" is the obviousness/necessity of a statement following a definitional statement?

Whether "bachelors are unmarried men" is clearly not necessary a priori in the sense that a system need not contain "bachelors", "marriage", or "men". In the same way, the way we define "12" is not necessary a priori.

Is your suggestion that because the universal quantifier may be necessary a priori, that the addition of "all" to the definition "bachelors are unmarried men" is analytic? Are you further suggesting that because the child need know the concept of "12" prior to realizing that "7+5" equals 12, that there is no parellel sentence "all 7+5's are 12"?
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keda
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Posted 05/27/09 - 04:20 PM:
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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.
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worldlogicleague
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Posted 05/27/09 - 05:27 PM:
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xzJoel wrote:


Are you saying that what makes something "analytic" is the obviousness/necessity of a statement following a definitional statement?

Whether "bachelors are unmarried men" is clearly not necessary a priori in the sense that a system need not contain "bachelors", "marriage", or "men". In the same way, the way we define "12" is not necessary a priori.

Is your suggestion that because the universal quantifier may be necessary a priori, that the addition of "all" to the definition "bachelors are unmarried men" is analytic? Are you further suggesting that because the child need know the concept of "12" prior to realizing that "7+5" equals 12, that there is no parellel sentence "all 7+5's are 12"?


That about sums up this angle. 12 doesn't belong necessarily and explicitly in the definition of "7+5" (as oppsed to the consequence of doing what it names), The analysis of the concept into its defining genus (additive operations) and differentia (where first is 7, and 2nd is 5), its definition, doesn't require 12 - it isn't contained in the subject concept.

keda above said it best -

"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."

Edited by worldlogicleague on 05/27/09 - 05:53 PM
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