# Every Scientific Hypothesis is a Hypothesis on a Finite Sample Space

From the p-value debate at Andrew Gelman's blog:

For every measurement of the macroscopic universe that will ever be performed by humans, that measurement will have some resolving power. Let’s pretend that the highest resolving power will come from an electronic A/D converter of some measurement instrument yet to be devised, and that it has 256 bits of resolving power. Today, very high quality A/D converters have maybe 31 bits

http://www.ti.com/lit/ds/symlink/ads1281.pdf

So under this hypothesis that the best ever measurement instrument will have 256 bits of resolving power, any scientific hypothesis involving sample spaces larger than 2^256 different possible finite outcomes is not a testable scientific hypothesis. PERIOD.

Now, let’s examine some physical reality of the universe: according to Wikipedia current approximate calculations give the number of protons/electrons in the universe as around 10^80

This means to have my hypothetical 256 bit A/D converter we would have to accurately count all the electrons in approximately 1/1000 of the entire universe. I assert that this will never occur, so every probability sample space on scientific measurements has less than 2^256 distinct discrete possible outcomes, each distinct outcome has a perfectly ordinary probability associated to it.

Continuous probability distributions are purely convenience for not having to work with an exactly known quantity of discrete outcomes, and not having to carry around sums that contain 2^15 terms and soforth.

Perhaps I'm wrong, perhaps some day we'll have a measurement which is accurate to 1 part in 2^257, the argument still survives. The exact number doesn't matter, its some big finite number. This is exactly why I love the nonstandard analysis of Edward Nelson (IST) because it's all about assigning a predicate to "bigger than we'll ever actually define, but still finite". Integrals are big finite sums, very big, extremely big, bigger than you'll ever actually define if you write towers of for the rest of the existence of the human race. It's big enough eventually.

Daniel,

I read your stuff all the time, but seemingly never get around to commenting. We have somewhat similar backgrounds, which differ from normal stat folks (me: separate master's degrees in Physics, pure Math, and Statistics).

You're exactly right about the inherently discrete nature of statistics. Although I had a long and intimate acquaintance with measure theory before statistics, I think E. T. Jaynes assessment was right:

(1) Measure theory isn't a more rigorous approach to statistics, it's just more general.

(2) The increased generality is completely irrelevant for applications.

Like many coming to statistics from a pure math background, I thought measure theory was the royal road to statistical insight. The exact opposite was true. I have yet to encounter a single instance in which measure theory even came close to solving any foundational problems with the philosophy of statistics.

-joseph

P.S. I spent some time back in the day on Non Standard Analysis. I'll get around to adding my $.02 worth at some point. It's a really obscure topic to most people.

P.P.S. In your Continuum Mechanics related work, do you ever run across the name Clifford Truesdell? I'm curious to know if his influence, such as it was, is completely lost.

Hi Joseph,

Thanks for your comment. I read your comments on Gelman's blog all the time. I feel like we have "kindred spirits" when it comes to philosophical issues in science and esp. stats. Glad to hear I have at least one reader :-). My own background was an undergraduate degree in mathematics with a minor in computer science (and another informal minor in Philosophy), then about 5 years applied stuff in finance, biotech, and statistical data analysis, followed by a return to UC Davis for a CE bachelors, a few years in Forensic Engineering, and then back to the PhD program here. I think the mix of applied and academic work has given me a kind of insider-outsider dual perspective on the usual academics. It's on of the things I like about Gelman for example is that he's really done a LOT of actual applied stats, so he knows about the *science* not just math and methods.

I haven't come by Truesdell's name, but I have mostly focused on recent developments in Continuum theories. For example I am writing a paper now on how NSA can help cut through certain mathematical issues and clarify the physical relationship between various types of continuum models, including the difference between the classical Cauchy Continuum, and Peridynamics, a recent continuum model developed by guys at Sandia which is extremely successful for fracture modeling. I'll send you a draft in the next few weeks if you're interested.

Using my NSA approach I came up with a model for dissipation of waves at the micro scale, which I tested against molecular dynamics simulations. It involved a lot of Bayesian fitting of parameters in the exploratory stage. I want to put this stuff up on my blog soon, but need to get the papers more fleshed out first.

It would be great to see what you’re working on.

I'm somewhat jealous of your Ph.D. work. I've been seriously thinking about going back and doing something applied/continuum mechanics related. Perhaps Geophysics or Meteorology. Truesdell lead a continuum mechanics revival about mid-century. His books are hard to come by now and are extremely expensive, but there’s nothing like them anywhere. His “Non Linear Field Theories of Mechanics” is available though. Years ago I took a few graduate Engineering Mechanics course because the coverage of that stuff in the physics graduate program was absolutely embarrassing. Seriously, the physicists were teaching this stuff at a level that would have been considered unacceptable in 1850. The textbooks that they used in the engineering mechanics courses made no mention of Truesdell, but the notation, vocabulary, and emphasis was clearly came from his school’s work. So although he seems to have had no effect on physics, others must have been paying attention.

Having done applied work seems to be much more important Statistics than in most fields. No current Philosophy of Statistics survives first contact with real problems.