As part of my Cosmology, Consciousness, and Computation course I have lately been thinking a lot about Zeno’s paradoxes and the ‘standard solution’ to them that we get from calculus and thinking about motion as a ‘completed infinity’, limits, etc (next week we start quantum mechanics and I am starting to wonder about how Zeno’s paradoxes might relate or not to that, but one thing at a time!). Take the function g(x)=1/x, as x approaches zero the function trends towards infinity (both positive and negative infinity depending on the direction one is going in). We put this by saying that the function’s limit is ∞ (in both directions), but ‘infinity’ is not a real number. So a different way of putting this point would be to say that there is no real number that is the limit of this function.
This got me to thinking. There may be no real number but does that mean that there is no number at all? Suppose that we introduced them on the lines of imaginary numbers. In homage to this let us call them ‘pretend numbers’. 1/0 will then be equal to some pretend number. I will use ‘Þ’ for pretend numbers, so ‘pretend one’ will be ‘Þ1‘. What is Þ1? It is the pretend number that when multiplied by real 0 will give you real 1. We can then define n/0 as follows:
where N≠0,
N⁄0=Þn,
N⁄Þn=0,
Þnx0=N.
These numbers are not on the real number line, or on the imaginary extension of that line. They are on a further pretend extension. Admittedly 0/0 is still tricky and I am not sure what to say about it but putting that to one side, do pretend numbers exist?
Richard Brown 
Which numbers do exist? Where, and in what sense?
very few numbers actually have a finite decimal representation, so why is say 1/3 more real than your guy (or sqrt(2) if you must), if we think of our ability to oprerationalize mapping the concept onto objects.
Why does it make sense to say that 0 exists, or -1?
But for that matter in what sense can we say that 1 (a unit) exists? it’s a precept. my body is “1” only if you happen to have the right kind of sensory/measurement apparatus in terms of spatial and temporal grain. Had you eyes the size of a few molecules would you have any reason to think that my oneness exists?
Are individual particles less ambiguous? maybe only sub-particles, but then we have position/momentum uncertainty and such.(let alone if string theory is true).
numbers are mental constructs, as is everything we will ever chance upon. as such they are as real as anything else (my justified vast set of beliefs non-withstanding).
I think the unbiased outlook here is usefulness and robustness. So the question is if your definitions actually help out (in proving a theorems, or via application).
Hi Richard,
Well, I suppose that in order to answer your question one would have to know more about their purported properties, how they’re supposed to interact with each other and with other numbers. For example, is Þ1 + 1 equal to Þ2? Or is it rather that Þ1 + Þ1 = Þ2? If the latter, then what *is* Þ1+1 equal to? Would it make sense to raise numbers (pretend or otherwise) to pretend powers? Also, other numbers can in various ways be constructed out of sets, or at least associated with them. Could the pretends (the term ‘pretends’ being used in analogy with ‘reals’) be associated with sets? Unless questions like this can be answered, I don’t think the question of the pretends’ existence have been given sufficient meaning to have an answer of its own.
Two further remarks:
First, since Þn x 0 = N, Þn/N would seem to be 0, and so Þn/0 would seem to be N. So Þn/0 = Þn x 0, maxing 0 analogous to 1 for pretends in the sense that multiplication and division yield exactly the same result. However, there is an oddity: If Þ1/0=1, then it seems Þ1/1 =0. Now, one would think that Þ1 x 1 = Þ1, and hence that Þ1/1 should be Þ1 and Þ1/Þ1 should be 1. We can’t have Þ1/1 be both 0 and 1, so something has to give. Either Þ1 x 1 equals something other than Þ1, or Þ1/1 equals something other than 0.(One might say that Þ1 x 1 = 0, which avoids our difficulty, but then what of Þ1/Þ1? A natural suggestion is that Þ1/Þ1 = Þ1, and hence that Þ1 x Þ1 = Þ1. This would imply that 2/Þ1 =(Þ1 x 2) x 0, I write it like this because Þ1 x 2 is presumably Þ2, and Þ2 x 0 = 2. Here multiplication is not associative:
Þ1 x (2 x 0) (with the parentheses in different places), since (2 x 0) = 0, is itself equal to 1.
Second, if the above difficulty can be solved, since one can divide imaginary and complex numbers by 0 as well, also have numbers of the form Þi (which is presumably the square root of Þ-1), Þ(ia+b), and so on. In short, one can extend not merely the real numbers but also the complex numbers.
There are several consistent ways to extend the real numbers with ‘infinite’ elements, the simplest one being just the projective extension (http://en.wikipedia.org/wiki/Projectively_extended_real_numbers), which one may view as a kind of stereographic mapping of the real number line onto a circle, with the ‘north pole’ being the infinite number. The downside here is that these projectively extended reals no longer form a field.
As for your pretend numbers, the problem is that they’re not consistent:
N = Þn*0 = Þn*(0 + 0) = Þn*0 + Þn*0 = N + N,
which is of course false for general N (that is, all N except N = 0). So I’d say they don’t exist.
You can get around this difficulty by dropping your third axiom, that N = Þn*0. This follows from N/0 = Þn only if 0/0 = 1, which need not be the case. But then, it can be proved that all pretend numbers actually must be the same.
To do so, I’ll change notation a bit, and write every number simply as a fraction n/m; the pretend numbers are then just those with m = 0. Addition of two numbers then becomes n/m + k/l = (nl + mk)/ml; this will be pretend if either summand is pretend. Now consider that we would want that n/m = jn/jm, where j isn’t 0 (such that we know that j/j = 1). But then, we would want the following to hold:
n/m + k/l = jn/jm + k/l, i.e.
(nl + mk)/ml = (jnl + jmk)/jml
But if now k/l is pretend, i.e. l = 0, then we get that mk/0 = jmk/0; since j is arbitrary, any two pretend numbers must be the same. And then, you’ve effectively only added one new infinite element to the real numbers, ending up with something like the projectively extended reals above.
As for Zeno, there’s actually some very juicy quantum mechanics associated with that name, because in the quantum world, a watched pot (truly) never boils: the quantum Zeno effect (http://en.wikipedia.org/wiki/Quantum_zeno_effect) denotes a situation in which repeated observation effectively ‘freezes’ the dynamics of a system. A full explanation would be a bit long for this comment, but roughly, upon an observation, the system is projected into some eigenstate of the observable; then, if the observation is repeated almost immediately, it will again be projected into that same eigenstate with high probability, such that a chain of such observations will effectively lock it in this state.
And it seems to be a perfect fit for a course about ‘Cosmology, Consciousness, and Computation’, because who first wrote about this effect? Alan Turing, that’s who.
Hi Jochen,
Interesting!
You say:
“As for your pretend numbers, the problem is that they’re not consistent:
N = Þn*0 = Þn*(0 + 0) = Þn*0 + Þn*0 = N + N,
which is of course false for general N (that is, all N except N = 0). So I’d say they don’t exist.”
But one thing Richard could say is that this just shows that the distributive law doesn’t hold for pretends. That is, we’d have:
Þn*0 = Þn*(0 + 0),
but *not*:
Þn*(0 + 0) = Þn*0 + Þn*0.
Would that lead to outright inconsistency?
Well, the distributivity for the pretends really follows directly from the distributivity for the reals; considering three numbers a/b, c/d, e/f, we have:
a/b*(c/d + e/f) = a/b*((cf + de)/df) = (acf + ade)/bdf,
and
a/b*(c/d + e/f) = ac/bd + ae/bf = (acf + ade)/bdf, so
a/b*(c/d + e/f) = ac/bd + ae/bf.
So monkeying with that would entail mucking about with addition and multiplication, and I’m frankly not sure we’d get anything sensible out of this—not to mention anything we could sensibly call an extension of the real numbers. But really, I think the fact that all pretends collapse to one already throws a spanner into the works.
What if we didn’t have numbers noted Þn, where Þn * 0 = 64, we could just assume there is the number Þ which is equal to 1/0? and when multiplied by 0, we allow the zeros to cancel (since our intent is to be able to work with zero in the denominator) and give us 1, or if Þ has a coefficient, the answer is the coefficient. this means that Þ + Þ = 2Þ, which means (1/0) + (1/0) = (1+1)/0 = (2/0) = 2*(1/0) = 2Þ. does this lead to contradictions that need to be addressed?
This looks suspiciously like something we covered in a graduate school topology class. Check out “single point compactification” if you’re interested.
There’s also the concept of negative absolute temperature in physics for a “physical” version of this. In that case, 1/T is the more natural quantity to consider (T= absolute temperature), and +/- infinite temperature are well defined (and very close to each other in value).