A seemingly simple problem like 2+2 turned io a paradox when real-world coexts came io play, and this sparked Kareem Carr’s discussions about the definitions and principles of the subject. Karim Karr shows how the coext can distort the calculation; For example, angles that return to 2π radians or rounding that can turn 2.3+2.3 io 5. The idea extends to statistics and politics, warning that scores like IQ or measuring emotions can be misleading.
On paper, this is one of the easiest math problems in the world. If you’re couing something like screws in a hardware store, the answer is pretty straightforward. But in other coexts, the boundaries are blurred. If you combine 0.48 liters of vinegar with 0.48 liters of baking soda and the resulting reaction produces 1.2 liters of foamy liquid, does that mean that 2+2 equals 5?
We iroduce assumptions io the world of mathematics. In this case, simple couing numbers, i.e. iegers 1, 2, 3 and the like, show a gap between mathematical abstraction and its application. Using the proposition 2+2=4 as food for thought, mathematicians are exploring situations where 2+2 does not actually equal 4, at least not in a regular way. These ierpretations can be extended to larger questions in epistemology, that is, how do we actually know what we know!
A few years ago, Karim Kar, a PhD stude in biostatistics at Harvard University, ignited a debate on the social network X about whether 2+2 can equal 5.
In his tweet, he emphasized that couing numbers are abstractions of underlying real things in the world, and as such, we must be careful how these abstractions distort the truth when iroduced io real-world scenarios. Arithmetic works well in the textbook, but in practice it is often faced with coextual questions that do not consider parts of a whole, approximations, or more releva vectors.
For example, if you add whole degrees to an angle, you end up with an angle that measures 2π radians. But an angle with a size of 2π radians has the same orieation as an angle with a size of 0 radians, so whether the angle is 0 radians or 2π radians depends on the coext. Similarly, if you turn a screw five full turns instead of four full turns, that is, 8π radians versus 10π radians, the orieation of the screw remains the same, but in one case the screw is driven deeper io the wood.
Replies to Kareem Carr’s tweet showed other examples of real-world accou restrictions. Many poied out that two animals could become three through breeding, or even one depending on the parameters, or that two machines could be turned io three machines using spare parts each and a little manual labor. Others noted that 2.3 rounds to 2, but 2.3 + 2.3 rounds to 5, and so under a certain filter, 2 + 2 = 5.
In general, the notion that we innately learn couing numbers, that is, only iegers and not fractions or decimals, is a common misconception among people not trained in mathematics or human developme. Young children learn numbers one by one through couing, but only begin to learn more advanced couing, larger numbers, when they can recognize quaities quickly. For example, we find it easier to cou to 7 when we can immediately recognize a group of four and then cou the fifth, sixth, and seveh. Couing is a learned and unnatural skill; Even non-human animals that can cou to four or five, such as dogs and chimpanzees. Therefore, the imposition of abstract numerical numbers on the real world creates an inhere tension.
