Mathematics

Suppose there is an infinitely long ladder in front of me. I do not know that this ladder is infinitely long, only that it is either a very long (but finitely long) ladder, or an infinitely long ladder. What kind of evidence would I need to give me reasonable assurance (I don't need absolute certainty) that this ladder is indeed infinitely long? I could walk a mile along the ladder and see that it still shows no signs of stopping soon. But the finitely long ladder would still be a better hypothesis in this case, because it explains the same data with a more conservative hypothesis. If I walk two miles, the finitely long hypothesis is still better for the same reasons. No matter what test I perform, the finitely long hypothesis will still better explain the results. Does this mean that, even if infinite objects exist, empirical evidence will never provide reasonable assurance that they exist?

Comments

In a finite lifetime, you won't be able fully to inspect an object with parts that are infinitely far from you, at least if we assume that you are limited by the speed of light. But there's other evidence. For example, you may be able to measure the gravitational pull of the ladder. If this pull turns out to be exactly what our theory would predict for a ladder that's like the piece of it we have before us (same material, thickness, density, etc.) and infinitely long, then this would be evidence for infinite length. (Note here that the gravitational pull exerted by any one inch of ladder declines with the square of its distance from you. So no matter how long the ladder its, its gravitational pull will not be infinite.) It's also possible that the ladder is expanding (as our universe is), or perhaps contracting. In that case you get a nice Doppler effect: a transformation of light reaching you from distant parts of the ladder -- the farther the light has traveled, the more strongly transformed it arrives. So evidence can provide reasonable assurance. And we have such reasonable assurance now that the universe is in fact finite.

But it's worth noting, as you suggest, that such evidence depends on "reasonable" assumptions. We assume that the laws of nature we've found to hold true around our space-time location also hold millions of light years away and millions of years in the past. Do we have reasonable evidence for this assumption? Well, yes, sort of, if with this assumption we can fit all our observations into one coherent account. Physics hasn't quite achieved this yet. But once physics delivers such a comprehensive theory, then it'll strike us as unreasonable -- bizarre -- to defend an infinite size of the universe by appeal to diverse laws of nature holding at different regions of space-time. But you should ask: the fact that some hypothesis strikes us as bizarre, how much reassurance is this that it's actually false?

In relation to my earlier answer, the following article from the Economist may be of interest. It's advertised as follows: "Can the laws of physics change? Curious results from the outer reaches of the universe." The link is

www.economist.com/node/16941123?story_id=16941123&fsrc=nlw|hig|09-02-2010|editors_highlights

This is not exactly what I had in mind, but relevant nonetheless.

BTW, this question is probably best classified under "physics" rather than "mathematics."

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