Example on pg. 23 of Mathematics for the Million

[bekkilyn]

I'm trying to figure out this example on page 23 of the book Mathematics for the Million by Lancelot Hogben. (Here is a link to the online book: https://archive.org/details/HogbenMathematicsForTheMillion/page/n25 )

While I understand the concept that adding two and two does not always equal four in the "real world" as the book states, I'm stuck on how the "law of addition" works for the particular figure displayed.

The example says the laws of addition for filling the figure with water would be: 1+1=2; 1+2=3; 1+3=2; 2+2=2.

Since it looks like the water could only go up to the line marked with a 2 before it would start leaking out through the pipe, so it seems like everything that doesn't result in a 1 would have a 2 result. I'm not clear on how 1+2=3, unless it has something to do with the way the pipe is curved, but if that was the case, wouldn't 1+3 and 2+2 also equal 3? But that's not the case either.

If it just means that you take the amount that would fill up the 1 partition and the amount that would fill up the 2 partition, then the result would be 2, and it would be impossible to get a 3 from any of these equations since they are at most only adding two partitions.

So why is 1+2=3 here?

 

[Mark44]

So why is 1+2=3 here?

Look closely at the figure. The water doesn't flow out of the tube until it gets to about 3-3/4 cups (or whatever the units are).

BTW, his example of how 2 + 2 might not equal 4 seems pretty stupid, to me. If you have a container that can hold 2 liters, and it already has 1 liter in it, if you add two more liters, you will get a total of 3 liters, but 1 liter will overflow out of the container.

 

[bekkilyn]

Look closely at the figure. The water doesn't flow out of the tube until it gets to about 3-3/4 cups (or whatever the units are).

BTW, his example of how 2 + 2 might not equal 4 seems pretty stupid, to me. If you have a container that can hold 2 liters, and it already has 1 liter in it, if you add two more liters, you will get a total of 3 liters, but 1 liter will overflow out of the container.

Right, but then why would 2 + 2 not also equal the 3-3/4 units instead of only 2?

 

[Mark44]

Right, but then why would 2 + 2 not also equal the 3-3/4 units instead of only 2?

My guess is because of the siphon effect. Clearly 1 + 2 = 3, but 1 + 3 puts the level above the top of the outlet pipe, so anything above the 2 level would be drawn out by siphoning.

In any case, these are silly examples of why 2 + 2 might not equal 4. My two cents.

 

[fresh_42]

I guess the idea is that the overflow starts at 2, which makes 1+2=2 not 3 in your first post.

Anyway, the idea is that addition is not defined in itself, it requires an environment, where it makes sense. A better example is perhaps a light switch. It is either up or down, and if you switch it twice you are at the null position again. Or a watch: after 12 hours it starts counting with 1 again: 6+8=2. So the operation "what do we mean by addition" defines the rules, not the plus sign.

 

[Mark44]

I guess the idea is that the overflow starts at 2, which makes 1+2=2 not 3 in your first post.

I don't think so. The container won't start overflowing until the level gets to about 3.75 or so. When it reaches this level, it will drain the level down to 2.

 

[jedishrfu]

He was using physics to convey an idea of when 2+2 =/= 4.

In deference to Hogben, you have to realize that his book inspired many folks to pursue math and science.

https://en.wikipedia.org/wiki/Lancelot_Hogben

https://www.britannica.com/biography/Lancelot-Thomas-Hogben

http://broughttolife.sciencemuseum.org.uk/broughttolife/people/lancelothogben

https://www.nytimes.com/1975/08/23/archives/lancelot-hogben-dead-popularizer-of-science.html

There's an interesting physics/math book that extends the concepts using physics to "prove" math concepts.

https://www.amazon.com/dp/0691154562/?tag=pfamazon01-20

 

[member 587159]

BTW, his example of how 2 + 2 might not equal 4 seems pretty stupid, to me. If you have a container that can hold 2 liters, and it already has 1 liter in it, if you add two more liters, you will get a total of 3 liters, but 1 liter will overflow out of the container.

I agree. The example is stupid and even confusing. The examples @fresh_42 gave in #5 convey the message much more clearly imo.

 

[jedishrfu]

You guys are looking at his work in the context of today's learning schemes. He wrote the book in 1936 and folks would have understood his example better than some chemistry example as in mixing two liquids of differing densities.

https://www.departments.bucknell.edu/chem_eng/eLEAPS/cheg200/DVofMixing/DVmix.pdf

Other schemes to show that 2+2 =/= 4 are changing bases like using base 4 so that ## 2 + 2 = 10_{base (4)}##

 

[bekkilyn]

I suppose what is really bothering me about this example is the inconsistency of the solutions.

It seems to me that the solutions would either be: 1+1=2; 1+2=2; 1+3=2; 2+2=2 or 1+1=2; 1+2=3; 1+3=3; 2+2=3 (or some other consistent result) but not a mixture of the two.

 

[jedishrfu]

But that's the point, the real world can do things that are unexpected. Imagine this bottle in a black box where you have a means to see the water level but no way to see how it works. You do your experiments and discover that it almost follows addition as we understand it but then there's a gotcha when you get to 2+2 or 1+3 or 0+4 and you wind up with 2.

Erno Rubik designed his cube and challenged his students to figure out how it works. Many would come up with elaborate schemes and then he show them the simple three axis scheme with interlocking pieces that both rotate and stay together.

https://www.cnn.com/2012/10/10/tech/rubiks-cube-inventor/index.html

Veritaseum presented some physics puzzles and challenged his viewers to figure out how they worked:

Spoiler: solutions

Bottomline, while arithmetic is a powerful tool for us to use, some things don't always follow the simple rules of arithmetic.

 

[bekkilyn]

I suppose it could be modeling an example of a large chunk of sediment getting into the water and eventually blocking the tube, so that 1 + 2 = 3 after the blockage, though before the blockage, 1 + 2 = 2 (assuming that the water normally drains down to 2 when unblocked).

 

[Mark44]

You guys are looking at his work in the context of today's learning schemes.

Well, I'm not. Hogben is talking about the operation of addition, and his explanation that 1 + 3 = 2 is silly. All together you end up with 4 units (cups? pints? whatever) of liquid, but some of it has run out of the container.

He wrote the book in 1936 and folks would have understood his example better than some chemistry example as in mixing two liquids of differing densities.

Being written in 1936 is no excuse

It seems to me that the solutions would either be: 1+1=2; 1+2=2; 1+3=2; 2+2=2 or 1+1=2; 1+2=3; 1+3=3; 2+2=3 (or some other consistent result) but not a mixture of the two.

1 + 1 = 2 -- yes, 1 + 2 = 2 -- no, 1 + 3 = 2 -- yes, for the reason I already explained.
If you add enough water to get the level above the high point of the external tube, the fluid will overflow and the siphon effect will take more with it, down to the level of the 2 mark. This is the only explanation that makes sense, as far as I can see.

Have you ever siphoned gas out of a gas tank? It's the same idea.

But that's the point, the real world can do things that are unexpected.

But this is just a parlor trick -- it isn't mathematics.

 

[bekkilyn]

Well, I'm not. Hogben is talking about the operation of addition, and his explanation that 1 + 3 = 2 is silly. All together you end up with 4 units (cups? pints? whatever) of liquid, but some of it has run out of the container.
Being written in 1936 is no excuse
1 + 1 = 2 -- yes, 1 + 2 = 2 -- no, 1 + 3 = 2 -- yes, for the reason I already explained.
If you add enough water to get the level above the high point of the external tube, the fluid will overflow and the siphon effect will take more with it, down to the level of the 2 mark. This is the only explanation that makes sense, as far as I can see.

Have you ever siphoned gas out of a gas tank? It's the same idea.
But this is just a parlor trick -- it isn't mathematics.

I think what you said above finally just clicked in my brain! It's because the siphon effect hasn't kicked in yet that 1 + 2 = 3. But 1 + 3 would equal 2 because the siphon effect kicks in around 3.75 or so, and 2 + 2 = 2 for the same reason.

Though I vaguely remember siphoning a gas tank in the past, it's not typically something I do a lot of in the "real world" so it was difficult for me to imagine the effect.

 

[jedishrfu]

@Mark44 I can appreciate your argument but will agree to disagree with it.

If we consider when the book was written circa 1936 and consider how we can provide an understandable example of when 2+2=/=4 to our readers then what would be the best real-world example for an author to present?

 

[Mark44]

@Mark44 I can appreciate your argument but will agree to disagree with it.

If we consider when the book was written circa 1936 and consider how we can provide an understandable example of when 2+2=/=4 to our readers then what would be the best real-world example for an author to present?

Again, the fact that it was written in 1936 is irrelevant to me. Why is it important to show that 2 + 2 != 4, except possibly as a clever parlor trick? The only halfway reasonable example I can come up with of why 2 + 2 might not equal 4, is arithmetic modulo 3, which was known about well before 1936, as far as I know. However, it's probably a stretch to consider that a "real-world example."

 

[PeroK]

I'm trying to figure out this example on page 23 of the book Mathematics for the Million by Lancelot Hogben. (Here is a link to the online book: https://archive.org/details/HogbenMathematicsForTheMillion/page/n25 )

While I understand the concept that adding two and two does not always equal four in the "real world" as the book states, I'm stuck on how the "law of addition" works for the particular figure displayed.

The example says the laws of addition for filling the figure with water would be: 1+1=2; 1+2=3; 1+3=2; 2+2=2.

Since it looks like the water could only go up to the line marked with a 2 before it would start leaking out through the pipe, so it seems like everything that doesn't result in a 1 would have a 2 result. I'm not clear on how 1+2=3, unless it has something to do with the way the pipe is curved, but if that was the case, wouldn't 1+3 and 2+2 also equal 3? But that's not the case either.

If it just means that you take the amount that would fill up the 1 partition and the amount that would fill up the 2 partition, then the result would be 2, and it would be impossible to get a 3 from any of these equations since they are at most only adding two partitions.

So why is 1+2=3 here?

I'd say that ##1 + 2 \approx 2.95## in this case, as some of the water goes into the syphon.

Why not just have an cone-shaped container? The horizontal lines would be at regular heights but the container would hold more as it widened as the water level rose. That would mess up the arithmetic as well.

 

[jedishrfu]

Here’s an example of how to use the glass cylinder in a water clock: