No Harvard, 2+2 Does Not Equal 5

With each passing year, it seems our institutions of higher education become less and less sane.

A recent, and egregious, example of this would be a controversial Twitter thread posted by Harvard PhD student Kareem Carr.

The student’s arguments were incompletely summarized by an article in Popular Mechanics (you tell me: is journalism dying or is journalism dead?) — a media mention which Harvard University itself was proud to feature on its website.

Carr’s basic thesis? Sometimes 2+2 = 5 — and it doesn’t get better from there.

Carr begins by arguing that basic mathematical statements are abstractions (which is true). But then he makes a series of statements that fly in the face of basic reality in the way only an academic can.

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Here, the only thing Carr manages to point out is that the appropriate application of mathematics changes over time to represent a reality. When there were a hen and a rooster, there were 2 animals. Then one year later there were 3.

Mathematics doesn’t suddenly break down in the face of this change. At the start, the formula 1+1 = 2 clearly applies; then one year later the correct formula to use is 1+1+1 = 3. At no point does 1+1 = 3. That’s just absurd.

He then uses this poorly formulated argument to come to an otherwise reasonable conclusion: “Whenever you create a numerical construct like IQ or an aggression score or a sentiment score, it’s important to remember that properties of this score might not mirror the real things being measured.”

This is a great point for any data scientist to keep in mind, but it has nothing to do with the nature and application of basic arithmetic and everything to do with the methodology and philosophy that goes into collecting and representing information.

If an IQ test is an inaccurate measure of intelligence, it is not because sometimes 2+2 = 5. If anything, its accuracy relies on the philosophy, methodology, and worldview that inspired the creation of the test and instructed its administration.

But perhaps I’m the wrong person to argue the other side. Perhaps Carr would more readily listen to a qualified academic such as David Berlinski, who holds a PhD in philosophy from Princeton University and was a postdoctoral fellow in mathematics and molecular biology at Columbia University.

In 2011, Berlinski published a book entitled One, Two, Three in which he outlines exactly why 2+2 = 4 every time.

At its core, it appears the foundation of Carr’s argument is the rejection of what many refer to as “natural numbers” (the idea that whole numbers are a naturally occurring reality and the symbols we use merely represent that reality). In fact, Popular Mechanics underscores this fundamental presumption from Carr when it refers to natural numbers as “a common fallacy among people who aren’t trained in math or, say, human development.”

Berlinski dedicates volumes to the theory behind natural numbers in the very first passages of his book — and he makes the argument with far more clarity and care than I could possibly write here. The following are just a few excerpts from One, Two, Three that are well worth the read.

“Were textbooks to disappear tomorrow, and with them the treasures that they contain, it would take centuries to rediscover the calculus, but only days to rediscover our debts, and with our debts, the numbers that express them.”

[…]

“If the natural numbers appear at the very beginning of human history, they will also appear spontaneously in every human mind. Otherwise, arithmetic could not be taught. Anthropologists are often amazed by the radically incommensurable way in which different societies organize the most basic facts of experience. Seeing this is said to be one of the pleasures of travel. Nonetheless, our own one, two, three, the Latin unus, duo, tres, and the Akkadian dis, min, es, designate precisely the same numbers. If goat eyes are a delicacy in the Khartoum but not in New York, it is nonetheless true that three goat eyes is more than two in both cities.”

“Because they are universal, the natural numbers very rarely are the cause of introspection. We take them for granted. We would be at a loss without them.”

[…]

“The natural numbers, Leopold Kronecker remarked, are a gift from God. Everything else is the work of man. This is a radical position in thought — an admission, on the one hand that the natural numbers cannot be explained, and a suggestion, on the other, that the mathematician’s proper work must be to accept this strange gift and from it derive all others.”

To assert that 2+2 can sometimes equal 5 does nothing to change reality; it serves only to distort and confuse the language we use to articulate and observe reality itself.

Carr is reportedly in the process of developing his graduate thesis on the basis of his ideas, and I sincerely hope that the final product is based on far more sturdy reasoning than he has so far expressed.