Can 1 x 1 be equal to 2?

Terrence D. Howard has recently challenged the accepted notion that 1 x 1 = 2 and has also offered an interesting proof in social media:

Anyone should be allowed to question common knowledge and I have no problem with that. Major discoveries in science were made when someone dared to question the status quo. An example is when Einstein challenged the common notion that the speed of light depends on relative motion and developed the relativity theory where the speed of light is the same in all inertial reference frames. That changed science.

Now, about 1 x 1 = 2.

Multiplication is not intuitive operation like addition because we can mix units. For example force is mass times acceleration (formally force is equal to rate of change of momentum).

F = m × a

What do we get when we multiply mass by acceleration?

The answer is we get kilograms times meters divided by seconds squared. That is a new quantity we call Newton and measures force.

So what is one Newton? It is equal to: 1 Kg × 1 meter/second^{2}

If we assert that 1 × 1 = 2 then we have to revise also whole of physics at least. This is fine as long as everything will remain consistent. But should we?

Let as take 2 × 2 for example. Everyone should agree that this is equal to 4. This is easy to confirm empirically. Two men each having two cars, there are four cars in total. In this sense

Number of cars = number of man × equal number of cars per man = 4

Note that the unit “man” cancels out as follows: man × cars/man = cars

Now, since we have to accept that 2 × 2 = 4 I hope because this is verified empirically, 1 × 1 = 1 is not that empirically evident, but the proof goes as follows:

Given: 2 × 2 = 4

Equivalent to (1+1) × (1+1) = 4

Distribution: ( 1 × 1 ) + ( 1 × 1 ) + ( 1 × 1 ) +( 1 × 1 ) =4

Equivalence: 4 × (1 ×1) = 4

Inverse element: (1/4) × 4 × (1 ×1) = 4 × (1/4)

Result: (1 ×1) = 1, necessarily, if 2 × 2 = 4

But we could assert 2 × 2 = 8, and get 1 ×1 = 2. But then things would get messy down the road. For example, we will be asked to call a 2 by 2 feet area an eight square feet area, or a 2 Kg times 2 meters per second squared, an 8 Newton force. I guess that would be fine if we all agree what we mean.

So I think to avoid any confusion and will stay with (1 ×1) = 1.

I am sure Giuseppe Peano, the amazing Italian mathematician, would suggest the same.

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