Mathematically Certain
We often come across the expression that a fact is “mathematically certain,” meaning it is undoubtedly certain because it has been validated by mathematics. This phrase has been used, for example, in discussions about the state of the U.S. economy after Trump’s first 100 days, or in Italy regarding employment and income. But does this expression really make sense? Do mathematically certain facts actually exist?
It’s worth taking a brief look at what mathematics is—and especially how it applies to real-world facts.
The certainty of mathematics lies in the fact that, from a logical standpoint, it is a tautology: a reasoning in which conclusions follow from definitions.
For example, if I define white as “that which is different from black,” I can conclude with absolute certainty that black is not white. Sure: if X is different from Y, then Y is not X. But this tells us nothing about what white or black actually are.
I could define a human as a being with two heads, and thus conclude that anyone with only one head is not a human. The reasoning would be absolutely logical and certain, but it wouldn’t apply to reality, because no such two-headed beings exist.
Now, coming back to mathematics, how do we prove that 2+3=5?
It is not an empirical observation (which wouldn’t guarantee certainty), but a result based on definitions found in rational arithmetic: adding two numbers means counting forward from the first by as many units as indicated by the second.
If we said 2+3=6 and not 5, we would fall into contradiction—but we would still be operating in a realm of meanings. Moving into the real world is something else entirely.
Similarly, in geometry, it has long been understood that nothing can be proven without starting from unprovable postulates. Hence we have Euclid’s famous postulate: “On a plane, through a point not on a given line, there passes exactly one line parallel to the given line.”
For thousands of years, no one questioned the certainty of mathematics. But at the end of the 19th century, new forms of mathematics emerged—non-Euclidean geometries—which describe spaces with a number of dimensions different from the three we perceive as the only possible ones. Today, there are mathematical systems with 4, 8, 12 dimensions, and so on.
Initially, these seemed like mere logical curiosities. But now, non-Euclidean mathematics is considered potentially applicable to the infinitely large or the infinitely small.
It is therefore necessary to verify, through experience, which mathematical systems are valid.
But even without venturing into the extremes of modern physics, it’s clear that traditional mathematics is applicable to the world of our everyday experience.
Precisely, “applicable.” But what does “applicable” mean? It involves applying abstract concepts (numbers, lines, points) to concrete facts, and that’s not simple or self-evident at all.
Let’s look at a common example taught in elementary school. We’re told that for a merchant, profit is calculated as:
profit = revenue - cost
From this formula, other relationships are derived.
But is it true? Not really.
If a merchant sells goods, they must buy them again—so profit depends not on how much was paid initially, but on the repurchase cost. In the case of price increases, one might think the profit goes up—but in fact, rising costs can reduce sales and thus reduce profits.
One must also consider that merchandise can spoil or go out of fashion. So, at some point, it might be convenient to sell it at a lower price than it was bought for. And there are many other factors to consider, like fixed costs, rent, employees, energy bills, and so on.
If we move on to physical applications, we find similar situations.
We might think that when pushing a broken-down car, four people exert twice the force of two. But if we extend this mathematical reasoning, we might conclude that twenty people exert ten times the force, or that two hundred people exert a hundred times the force. But this isn’t the case: beyond a certain number, people get in each other’s way. There is a maximum number beyond which effective cooperation becomes impossible—and this can’t be calculated mathematically, but only through experience.
Furthermore, if there are four people, perhaps one is only pretending to push. We can’t call that a mathematical fact.
This issue becomes even more complex in the realm of social and political facts.
We might (basically) know whether the number of migrants has increased or decreased—but does this tell us anything about whether the government in office deserves credit or blame? Or is it due to the many other variables involved?
In the same way, if employment, inflation, or GDP increases, is it because of government policies? There is no mathematical certainty here—these are facts that may have countless, intertwined causes.
When we apply mathematics to the trajectory of a missile, the variables are few and easily identifiable. But in the human sphere, they are infinite and difficult to assess or identify.
There will always be many interpretations to discuss—and no mathematical certainty.