# Semantic difficulty

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 1374042.  Tue Feb 09, 2021 12:28 pm Like a circle in a spiral, like a wheel within a wheel...

1374052.  Tue Feb 09, 2021 1:56 pm

 Brock wrote: OK, so you're saying that in the puzzle, the noun "bird" is being used as a synonym for the noun "ten" (i.e. a group of ten).

No, I'm not saying that it's a synonym for ten. It's a variable, which yields a true statement when it takes the value 10.

The trouble is that you seem to think that there is an important semantic distinction between
(i) "three x ten = 30", and
(ii) "three tens = 30";
and that in (i) "ten" means "the number 10" whereas in (ii) it means "a group of 10 objects" - and by extension, in (i) "30" is a number, whereas in (ii) "30" means "a group of 30 objects". In other words, you think that (ii) is exactly equivalent to
(iii) "three groups of ten make a group of 30".

On the other hand I (and I think most people on this forum) would regard (i) and (ii) as exactly equivalent, while (iii) is just how you try to teach children the concept of multiplication, as repeated addition.

Moreover, your interpretation only works over the natural numbers. What would you make of "ten fifths = 2"? You cannot say that "fifth" means "a group of fifth objects", or even "a fifth of a group of one object"!

 1374053.  Tue Feb 09, 2021 2:00 pm Succinctly put

1374054.  Tue Feb 09, 2021 2:38 pm

 Brock wrote: There doesn't seem to be any way of assigning a meaning to the puzzle that makes it coherent. And I'm saying that that's a semantic difficulty; three birds can't be equal to a number. They can have a value; but that's not the same thing.

I don't know how many people posted on the original discussion, but from replies on here, I'm getting the impression that it's coherent to most people, just not yourself, so perhaps it's more about your approach.

For example, you say above that a bird can be given a value, but not be a number.

However, the question then arises, what is a number?

To most of us on this site "10" or "Ten" symbolise a specific number. For other people that specific number is symbolised by something else, be it Eser, Asra, Tsen, Zece, Shi, or many others.

So why can't "Bird" symbolise the number 10?

Why does it have to be a separate value?

1374058.  Tue Feb 09, 2021 3:47 pm

 CB27 wrote: So why can't "Bird" symbolise the number 10?

It can. That was the whole reason why I talked about the use-mention distinction in my original post.

"Bird" is a symbol; it can represent the number 10.
A bird is not a symbol; it can't represent the number 10.

Cheese is not "cheese", and a bird is not the same thing as the symbol "bird". They're entirely different things.

However, I'm getting the feeling we're going round in circles, so unless anyone has anything new to say, I'll stop there. Thanks to everyone for contributing.

1374059.  Tue Feb 09, 2021 3:58 pm

 Brock wrote: I'll stop there.

Hoorah

1374060.  Tue Feb 09, 2021 4:06 pm

barbados wrote:
 Brock wrote: I'll stop there.

Hoorah

This is why I gave up formal semantics after two terms. It does your head in after a while.

1374097.  Wed Feb 10, 2021 2:31 am

Brock wrote:
barbados wrote:
 Brock wrote: I'll stop there.

Hoorah

This is why I gave up formal semantics after two terms. It does your head in after a while.

Perhaps you’ll recover eventually...

 1374098.  Wed Feb 10, 2021 2:35 am No sign of it so far... PDR

1374129.  Wed Feb 10, 2021 6:22 am

 CB27 wrote: I'm getting the impression that it's coherent to most people, just not yourself

To be fair to Brock, he's repeatedly stated that he understands the puzzle as written, he's simply indulging in nerdy over-examination of the semantics surrounding it. If we're going to ban nerdy over-examination from these forums, I'm really not sure what will be left to talk about.

 Brock wrote: So a bird is a group of ten, a cat is a group of three, and a dog is a group of one. A group of ten plus a group of three can form a group of thirteen; but what is a group of thirteen times a group of one? Nothing that I can make any sense of.

Building on what Numerophile wrote earlier about "three groups of ten make a group of 30", that surely answers your question.

A group of ten plus a group of three forms a group of thirteen (as you stated). Then a group of thirteen "times by" a group of one is the same as saying "one group of thirteen."

Or, of you prefer, "thirteen groups of one". Either way it's the same answer.

1374130.  Wed Feb 10, 2021 6:34 am

 Numerophile wrote: “3a = 30” is not an assignment, and neither, of course, is it a statement about the real world; it is an abstract statement, which is true when a is replaced by the appropriate value. The same applies to “Three birds = 30”. If “Three feathered animals = 30” doesn’t make sense, you need to look for an alternative interpretation that does. Since the right hand side is a number, the left hand side also needs to be a number, and when “bird” is replaced by “ten” it makes perfect sense; i.e. you solve the equation that results when you treat "bird" as an unknown variable.

I found this post particularly interesting. You state that "Since the right hand side is a number, the left hand side also needs to be a number". However, that's not necessarily the case, surely.

As you've pointed out, the mathematical "=" sign states an equivalence between the statements on either side. Throughout this whole discussion about "Three birds = 30", everyone seems to have assumed that the left hand side must match the right hand side (i.e. be a number). But the "=" sign works both ways.

Therefore it's equally valid (and in the spirit of nerdy over-examination of semantics) that "Three birds = 30" is actually stating that the symbol "30" is not being used as a number in this case but has been assigned a meaning to represent the concept of "Three birds".

Naturally the logical conclusion of this train of thought is that the mathematical problem becomes even more meaningless. This will probably score you more points from a philosophy or semantics teacher than a maths teacher, but it does remind me about the dangers of trying to write an overly simplistic question in the same way as the famous "Measure the height of a tower with a barometer" question :)

1374132.  Wed Feb 10, 2021 7:08 am

(Barbados had better go and hide now. Dr. bob has said something new, so I'm not breaking my promise of "unless anyone has anything new to say, I'll stop there".)

 dr.bob wrote: If we're going to ban nerdy over-examination from these forums, I'm really not sure what will be left to talk about.

Very well put!

 dr.bob wrote: I found this post particularly interesting. You state that "Since the right hand side is a number, the left hand side also needs to be a number". However, that's not necessarily the case, surely. As you've pointed out, the mathematical "=" sign states an equivalence between the statements on either side. Throughout this whole discussion about "Three birds = 30", everyone seems to have assumed that the left hand side must match the right hand side (i.e. be a number). But the "=" sign works both ways.

That's a very good point. When I first saw the problem, my initial reaction wasn't "oh, so 'bird' must represent a number in some way". It was "three birds equals thirty what"?

Had it been "three birds = thirty sheep", I think I'd have had far less of a problem. I'd have seen it as setting up an equivalence between birds and sheep in some way, so that a bird is equivalent to ten sheep. Even though birds and sheep are different creatures, I'd have taken it as meaning that a bird had the value of ten sheep in some way.

But "three birds = 30" seems like a grammatical anomaly, not just a semantic one. "Three birds" is a noun phrase, and I would expect the expression on the other side of the equals sign to be a noun phrase as well. The only way I can interpret "30" as a noun phrase is by taking it to represent the actual numeral "30"; but a numeral isn't a number, and you can't divide it by three (it's the use-mention distinction again, because "30" isn't the same as 30).

If anything has been established by this discussion, it's that in order to interpret the expression "three birds = 30" coherently, you have to change the usual meaning of something within it. Some people have proposed changing the meaning of "bird" so that it behaves like an algebraic variable; some people have suggested changing the meaning of the equality operator; and now you've suggested changing the meaning of "30". But you can't take all the parts of that expression with their usual meanings, and come up with a coherent statement.

1374142.  Wed Feb 10, 2021 8:49 am

 dr.bob wrote: Therefore it's equally valid (and in the spirit of nerdy over-examination of semantics) that "Three birds = 30" is actually stating that the symbol "30" is not being used as a number in this case but has been assigned a meaning to represent the concept of "Three birds".

Well, yes; and I am certainly open to any interpretation that makes "three birds = 30" coherent by changing the usual meaning of "30" - provided of course that it can be extended to "three cats = 9" and "three dogs = 3" as well, since that was the context in which the problem arose.

Any suggestions?

If not, I think we would have to rely on Friar William's shaving implement...

 Brock wrote: But you can't take all the parts of that expression with their usual meanings, and come up with a coherent statement.

Since nobody has either claimed or implied that you can, why do you bother to deny it?

1374144.  Wed Feb 10, 2021 9:13 am

Numerophile wrote:

 Brock wrote: But you can't take all the parts of that expression with their usual meanings, and come up with a coherent statement.

Since nobody has either claimed or implied that you can, why do you bother to deny it?

I'll remind you of the assertion I made in my original post:

 Quote: I continue to claim that "three birds = 30" is incoherent, on any reasonable interpretation of the words and symbols therein.

It all comes down to what you regard as "reasonable".

If you think that treating "bird" as an algebraic variable is reasonable, then it's not true. If you think that treating "=" as an assignment operator is reasonable, then it's not true. If you think that treating "30" as a special symbol for three birds is reasonable, then it's not true.

If you don't think any of those things are reasonable, then it is true.

1374310.  Fri Feb 12, 2021 9:28 am

 Numerophile wrote: Well, yes; and I am certainly open to any interpretation that makes "three birds = 30" coherent by changing the usual meaning of "30" - provided of course that it can be extended to "three cats = 9" and "three dogs = 3" as well, since that was the context in which the problem arose.

Of course it can. Currently we choose to arrange the letters "aceehrstt" in a particular way to represent the concept of a pair of cats standing next to another cat. So the problem can be interpreted as defining the number "9" to represent the same concept, as well as defining "3" to represent the concept of a trio of hounds.

 Numerophile wrote: If not, I think we would have to rely on Friar William's shaving implement...

Just because that interpretation makes the mathematics impossible doesn't mean it's a wrong answer. You could reply by saying one bird plus one cat equals a bird and a cat. Or possibly just a cat that's no longer hungry. You could then point out that it's impossible to multiply anything by a dog.

That is not a wrong answer.

In the same vein, if someone is asked "How would you measure the height of a tower using a barometer?", and they reply "I'd tie it to a long rope, lower it from the top of the tower, then measure the length of the rope (plus the length of the barometer)", that is also not an incorrect answer. It may not be the answer the question setter was expecting, but that says more about the question setter than the response.

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