The Latin word commensurabĭlis came to Castilian as commensurable. This is how what can be valued or measured is qualified. Instead, something that is not subject to a valuation or measurement is immeasurable.
According to DigoPaul, the commensurability is the condition of what is commensurable. In the realm of mathematics, two real numbers are commensurable when their ratio is a rational number. Let’s look at some of the concepts involved in this definition to better understand the concept.
First is the ratio, a relationship between magnitudes of two different sets, which is usually expressed in one of two ways: a: b or a is ab. When we speak specifically of numbers, the ratios can be expressed as a fraction (a / b) and, depending on the result, as a decimal number.
In this particular case, for two numbers to be considered commensurable, they must belong to the set of real numbers, that is, to the one in which both the rational (negative, zero and positive) and irrational numbers are found. Before going on to define irrational numbers, we must point out that for commensurability to be fulfilled in the field of mathematics, the result of the reason must be a rational number ; otherwise, if it is irrational, then we speak of incommensurability.
In the set of irrational numbers we find all those that cannot be expressed by a fraction a / b, where a and b are integers and b is not equal to zero. In other words, an irrational number is any real that is not rational, and that does not have an exact or periodic decimal expression.
Commensurability in mathematics does not focus only on the possibility of comparing numbers, but on the presence of a common factor that we can express. Its use originated in translations of the treatise on mathematics and geometry written by the Greek scientist Euclid around 300 BC, entitled Elements and composed of thirteen books.
Although Euclid used the concept of segment congruence instead of real numbers (for example, he developed an algorithm that today bears his name and is used to find the greatest common divisor), his theories and conclusions laid the foundations for current notions of commensurability.
All the products offered in a market are commensurate with money. When entering a clothing store, to cite one case, we can see pants that sell for 100 pesos, a jacket for 500 pesos, a bathing suit for 90 pesos and a shirt for 210 pesos. These garments (pants, jacket, bathing suit and shirt), therefore, are commensurate: they have a valuation, in this case economic. Taking this valuation into consideration, the potential buyer can finalize the operation or withdraw from it.
There are other issues that, on the other hand, are immeasurable since they cannot be measured or valued. An example is happiness. What is its value or its price? It is impossible to determine. Nor can it be argued that a person is 64% happy or that they have 42 happiness points.
It is common to find the idea that the truly valuable things in life are immeasurable, and that happiness, love and well-being enter into this set. Despite this, the literature admits the use of expressions such as “immeasurable love” or “immeasurable joy” to emphasize the depth and intensity of these feelings.
In the field of philosophy of science, theories can be commensurable or incommensurable depending on the existence or absence of a common theoretical language. When that language does not exist, theories cannot be compared and are therefore incommensurable.