Possibly, 7 isn't prime [closed]

Question

I'm curious if in the philosophy of mathematics (or perhaps the philosophy of modality), the following has been proposed:

There exists something like an imaginary (but not complex) number i such that i is not equal to 1 or 7 and yet i divides every positive integer n (including 7).

Without saying anything about whether such an i actually exists in this world, it seems conceivable* that such an i exists in some other world. From here it would follow that 7 isn't necessarily prime.

*The reason why such an assertion seems conceivable is that this is exactly the sort of assertion mathematicians have made for centuries about the existence of the complex number i that is the square root of -1.

Answer 1

Under the usual definitions, there is absolutely no possibility that 7 is a prime number, in this or any other universe.
The usual definitions are, say (among many equivalent ways of stating them), that

• a positive integer n is prime if n ≠ 1 and n has no positive divisors other than 1 and n, and
• an integer d is a divisor of an integer n if there exists some other integer c such that n = cd.
• (Plus some more definitions, in more detail, of "integers", of products, etc.)

So the fact that 7 is a prime number is simply stating that none of the numbers 2, 3, 4, 5, or 6 is a divisor of 7, which is true.

The existence of other possible worlds has no bearing on whether 7 is a prime number, because the "world" that is of interest to mathematics is completely defined by the mathematical axioms. (A question of philosophical interest is what relation this mathematical world has to our real world -- in another universe this mathematical world of integers may be less relevant to that universe -- but that is a separate question entirely, unrelated to whether 7 is a prime number or not.)

Note that even in the case of the square root of -1, there was no previously beleived assertion that was discovered to be false by the introduction of complex numbers: -1 is still not the square of any integer, nor of any rational number, nor of any real number, and will never be, and we can prove all of these with the usual axioms. It so happens that there exists an extension of the set of real numbers in which it happens to make sense to speak of a "square root" of -1: but to do so we must also extend the meaning of "square root" (the domain and range of this function).

Returning to the case of 7, there is a very simple alternative way to get a mathematical "world" in which 7 is not prime: simply move from the ring of integers to the ring of rational numbers. Here, it is possible to extend the definition of prime numbers (everything carries over) to show that 7 is not prime, because for instance it is the product of two numbers (3/2) and (14/3), to pick just one example. (In fact, in the ring of rational numbers, we can show there are no prime numbers at all, as every p/q can be written as the product of 2p/q and 1/2, say.)

As a less trivial example, consider the ring of numbers (a + b√-3), where a and b are both integers. The good old integers are a subset of these, got by taking b = 0. Here, though it may not be obvious at first, it so happens that 7 is not prime, because 7 = (2 + √-3)(2 - √-3). However, there are other prime numbers in this ring: you can prove that 5 is a prime number even in this bigger ring.

In either of these examples, nothing has changed about the properties of 7. Only the definition of "prime number" has changed, and it should not be surprising that with a different definition, things can change. However, in any parallel universe, in the ring of integers and under the usual definitions, 7 will continue to be a prime.

Answer 2

The answer to your question is "No", nobody has proposed such a number.

Because an infinite number of them exist.

0.1 is an example, so is 0.01, 0.2, etc etc etc.

As soon as you limit your "i" number to being a positive integer (and hence being relevant to the definition of prime), it becomes impossible, as there are a finite number of integers between 1 and 7, and all of them can be proven to not divide into 7. Anything bigger than 7 will obviously not divide into 7 with an integer result. Anything less than 1 is not a positive integer.

So although there are numbers that divide evenly (ie with an integer result) into 7, none of them (besides 1 and 7) are positive integers, and that's what it takes to make 7 prime.

When first conceiving "i" as a concept, it is important to realise that it is NOT part of the number systems you already know. Much like a child learning about fractions for the first time, or negative numbers, these are outside of the systems they have learned so far. It is possible to have a "new" concept that fits a definition that is currently impossible, but you cannot try to infer that it fits in an existing set of numbers (positive integers), and hence breaks an existing rule (prime).

Answer 3

I'm going to answer what I see as the intent of the question, rather than the question itself (which seems to be answered pretty well in the earlier answers). In short, I would say that the answer is yes - in the sense that people have wondered if there are meaningful extensions of the integers other than the Gaussian integers, and where I'm being a bit loose with what I mean by 'meaningful.' To give it more substance, I mean an associative algebra over the real numbers where every nonzero element is invertible. (You'd go really far by thinking of this intuitively as something that behaves like the integers, but maybe are a little different).

And there are.

There are also quaternions. These are 4-dimensional numbers in the same sense that complex numbers are 2-dimensional numbers, but with some more complicated multiplication rules. Further, the quaternion integers actually extend the Gaussian integers. What's more, $7$ does factor in the quaternions. For that matter, every 'natural prime' factors in the quaternions - quaternions have also been called the 'square roots of the primes,' and you can make this more rigorous. (But I won't)

But then again, quaternion multiplication isn't even commutative. So are they really like the integers? You could go higher, to the octonions, which are 8-dimensional. But they're not even associative. You could go higher still - there is a way - but you keep losing more and more of what you associate with the integers.

But these are it - there are no more. For a proof, I might direct you to this question at Math.SE, or to some background and intuition.

I'd also like to say that it's very challenging to decide what you mean by 'exists' anyway. Much of math is built on the same or a very similar set of axioms, and this is what I mean. But a very popular question is Do complex numbers exist?, which is nontrivial (I would say). Similarly though, you might ask in what sense natural numbers or negative numbers exist. I say this only because I'm sure it would be possible to come up with a different collection of axioms/a different idea of what is allowed to constitute 'a number' and have 7 not be a prime.

Answer 4

-1 is an integer having the property you specified. In general, you may be thinking of a unit in a ring. That's an invertible element.

Answer 5

I haven't checked this. But lets work in the simplest possible algebraic system possible. This is a monoid - that is we have an multiplication (not neccesarily associative - they usually are) and identity. Suppose there are seven elements. Define a linear order with the idenity as the first element and the last one denote as seven.

In this context the idea of a prime element still make sense.

Now fill in the rest of the multiplication stipulating only that every element must divide seven. Since the multiplication is not-associative this can be aribitary.

Hence we have an algebraic system which should satisfy your requirements.

(proviso: Of course this system has only the vaguest of relationships with the usual integers).

Answer 6

Is there a possible Counterpart theory way that we might make sense of this? Suppose that 7 in our world is the natural number 7, but in some other possible world, the proper counterpart of 7 is in some "Hypocomplex" field such that it has a square root. Taking a Lewisian interpretation of possibility, this would be the way to go to validate the claim that 7 is possibly not prime.

The issue then would be about deciding whether there was such a possible world, and whether it would be accurate to pick out a counterpart of 7 satisfying the appropriate relational properties.

There doesn't seem to be any problem in saying that there is a possible world in which there is a square root of negative 1. That's because the Complex numbers are effectively a way of describing pairs of reals, which can be done in set theory. Since it can be done in Set Theory, and the standard Kripke semantics of possible worlds is also set theoretic, what doubt can there be? So your strategy first needs a formal component; you need to give the semantics of the arithmetical language where your "quimaginary" numbers exist. That doesn't seem implausible, but you've got to do the work.

Then you're going to have to make an argument to the effect that you're not equivocating when you talk about 7 in this model as a counterpart to the 7 of the standard model of arithmetic. This argument is much more likely to fail, since we tend to associate prime numbers with the study of finite arithmetic, rather than with functions over the infinitary areas of maths like complex fields. Your counterpart relation is going to have to be very complicated, to the point where you probably have to give up on uniqueness or symmetry of counterparthood.