We write a number 15, or 26, or 42, or 196,884.* But that is only its decimal-system notation. Its symbol would look different in base 3 or base 8. E.g. decimal 15 is 1111 in the binary system. But the essence or reality of the number-in-itself remains the same, regardless of the symbolic system (binary, decimal, base 8, or other) in which it is expressed. It is like the real object (thing-in-itself, Ding-an-sich) and the word denoting it in different languages (German, French, or other). It is a relationship ofa symbol to its referent. The word “table” is not the object “table”. The map is not the territory. They belong to different worlds of meaning.

So what are the properties of the number-in-itself? First, it may be a prime or. a composite – a product of two or more primes, uniquely so defined. No two numbers have the same string of factors, just as no two human individuals have the same string of bases in their DNA. (Even more so for numbers, since the possible sequences of DNA bases, while extremely large, are finite, but the number of combinations of prime factors is infinite.)

Secondly, a number-in-itself may denote a geometric object or several of them. The dots on a die cube give examples up to 6. But a 3 could also be a triangle, or a 6 a hexagon. There can be space object, like 27 being a 3×3×3 cube. There are multiple geometric shapes for larger numbers; each may display the factors of composites, but the hexagon does not. If there are 4 or more prime factors, the object may have more than 3 dimensions, but it may also be a polygon or something in between. However, primes are not representable by simple regular shapes. Primes are like chemical elements, composites are like chemical compounds (e.g. H20 is an angle, but C02 is a straight line).

Positive integers are an infinite series of mathematical objects, each completely different from all others, like individualized (non-fungible) personalities, even though to human minds that personality is difficult to perceive when the number is large (easy for small numbers). Possibly a mathematical genius could discern the personality of large numbers. Most of us lack this faculty.

• The numbers 26 and 196,884 come from an article in scientific American, November 1998, pp. 40-41, about Richard Borcherds. The number 42 comes from “Hitchhiker’s Guide to the Galaxy”, as the computer’s answer to the meaning of the universe. (42 = 2×3×7, which means something in numerology.)