Originally published by Robert Beisert at fortcollinsprogram.robert-beisert.com

Questioning the Infinite

(I had a lot on my mind this weekend, so I took a brief hiatus from “Me Programming”. Enjoy a speculative theory, instead.)

We have a problem with the concept of infinity, as it is mathematically defined: it is not finite. It’s a question that has puzzled many a philosopher, and I wish to demonstrate the reasons why a philosopher would question whether such a thing is vaguely possible.

Observations

In the first place, we do not observe expansive infinity – that is, a size or quantity that is sufficiently massive as to be beyond any possible finite measure. An infinite number of photons (the smallest particles we can reasonably observe) in the universe could only be possible if we presupposed an infinite universe, else the universe would be impossibly stuffed with photons. And an infinite universe can never be observed, because we can only ever demonstrably prove boundaries (thus, finite space). Furthermore, all current observations of the universe have found no repetitions, which based on our observations and estimations of possible energy potentials is required in an infinite universe, so we have no evidence on which to found the assertion of an expansive infinity of space.

In the second place, we do not observe restrictive infinity – that is, a sufficiently small size or quantity that is beyond any possible finite measure. Currently physicists hold the smallest possible distance to be the planck length (roughly 1.6*10^-35 meters), and we have thus far never successfully captured an instant of time. We thus have no basis to posit any restrictive infinity in the natural world, as all observations have been of limited resolutions.

Thus, based solely on material observation and theory, we find no basis for a conception of infinity.

Mathematic and Philosophical Implications

Every school child knows (or should know) that the fraction 1/3 cannot be replicated using real numbers. It is an easily observable reality that an apple can be divided into three portions (though the exactness of that fractional division may be debatable), but in numeric representation it is not equal to 0.3, or 0.33, or 0.333, because three of those fractions does not reflect the whole (0.9, 0.99, 0.999). In point of fact, no matter how far you draw the finite scope of numbers (say, 0.9999999999999999999999999999999), it will never be equal to one whole.

To resolve this apparent issue, we turn to the concept of restrictive infinity. Because no number of finite divisions can yield the correct result, we suppose that we can produce an infinite set of increasingly tiny divisions such that the distance between the value 0.999…9 and 1.00…0 is infinitely small. We assume as well that, at the infinite boundary, there can be no numeric value of distance between 0.999…9 and 1.000…0, and thus the equation 1-0.999…9=0. By this reasoning, 1=0.999…9.

Using the principles employed by Zeno in his infamous paradoxes, we can derive the following problem. Assuming that the infinite regression of 0.999…9 can produce a value such that it is equal to 1, it can also be hypothesized that there is an infinite regression such that 1.00000…01 = 1. By this principle, a value can be infinitely close to 1 from either above or below. As this is not the case exclusively for the number 1, we can equally hypothesize that there is an infinite regression such that 0.9999…98 = 0.999…99 and 1.000…001 = 1.000…002. By the transitive property, then, we can determine that the infinite regressions 0.99999…998 = 1 and 1.000…002 = 1. Carrying out this process an infinite number of times, we can produce the absurd result that 0 = 1 = 2.

Of course, the reason why 0 != 1 != 2 is that there is distance between these values. That distance must come from the reality that there remains an extraordinarily small distance between even infinitely close values in our mathematic system.

Thus, the well-known set equation 1 – SUM[ lim(n=1, n->inf) 2^-n] = 0 must be false, or all of mathematics must become meaningless. Rather, it is right to say that the equation approaches, but never actually reaches, 0.

Thus, we fail to resolve Zeno’s paradox of Achilles and the Tortoise, even using calculus. The paradox – that Achilles can only ever approach the tortoise but never reach him because there is always another subdivision of space between him and the tortoise – seems mathematically sound.

Why is this Relevant?

Mathematicians are prone to any number of fallacies, the majority of which Zeno posited thousands of years ago. Even the physically impossible (and thus, unnatural) conception of infinity cannot resolve the dispute between numeric math and the physical reality. Three thirds resolves perfectly to one whole, but three of even infinitely detailed numeric depictions of one third (0.333…33) never actually resolves to one whole – there is a remainder, no matter how fine your resolution.

All errors propagate as the bounds expand. The sum of supposedly infinitely small values can be resolved to a whole (e.g. the sum of all “dimensionless” points on a line produces that line), demonstrating that there must be dimension in reality. I have yet found no way around this conclusion that dimension exists even at the finest imaginable resolutions.

The ramifications are interesting:

1. The universe cannot be reasonably assumed to be infinite, and the conceptions of parallel universes derived in part from that assumption cannot be relied upon

2. The natural world can only reasonably be assumed to be constructed of dimensioned quanta

3. Space and time must have dimension – there are no “instants” or “dimensionless points”

4. All models of reality must be defined within boundaries (including degree of reliable resolution), as the natural world itself appears to be bounded

5. Mathematics is not exempt from the boundary and resolution requirements of all models of reality

6. Any models of the Divine must be held within boundaries and resolution, as we ourselves are limited by boundaries and resolution requirements

7. At some point, there must be a sum total of all possible knowledge and information in the universe