For me, the purpose of doubt is to motivate inquiry. When any particular doubt no longer serves inquiry, I retire it.
If the purpose of doubt were to eliminate doubt, it would be far more efficient simply never to doubt.
Therefore, I doubt your philosophy of doubt. Let the inquiry continue.
When you wrote "But neither does it seem like the same shade of uncertainty" I suppose you mean that it doesn't seem that way, to you. Nor does it to me. But before, as a thinking person, I suggest that the difference is meaningful, I need a context or a reason. You haven't provided one, and that's why your argument has the flavor of religion, to my palette.
I'd love to see your answer to the actual skeptical argument, rather than the straw man you offer, here. Here you are doing the equivalent of announcing "I'm thinking of a number!..... 5!...... I'm right again! My quest for order is rewarded!"
If you use mathematics to find order in the messy world, and you succeed, does that amount to a proof that the order you found is the actual order? Kepler would have argued yes! So would have Newton. Both were wrong. We know they were wrong. Wrong but their ideas are enduringly useful, as far as we know... so far... The skeptical position is not one of denying the value of ideas, but rather that of continuing the inquiry.
When my inquiry ceases, my beliefs become hardened premises that define my world and prevents me from benefiting from ideas of people with different premises. That's fine in a simple world. A gamer's world. I've become convinced that there is no simple world, except in our fantasies. Overcoming bias is about finding our center in a messy world. It's about overcoming fantasy.
I love math. It's the only reason I sometimes wish I'd stayed in school. When I get rich, I want to hire a mathematician to live in my basement and tutor me. I bet they can be had for cheap.
Pure math is potentially a perfect idea. Applied math; not so much. When you see that line of 2's, how do you know it continues forever? You don't. You're making an induction; a beautiful guess. It's only because you peeked at the real answer-- an answer you yourself created-- that you can confidently say that you "predicted" the sequence with your method.
I'm much more interested in sequences produced in a simple deterministic way that are extremely difficult to crack. The move from "it makes no sense" to "it's obvious" is a critical dynamic in human thought. I'd like to see you write about that.
As Polya would say, solving these problems is a heuristic process. The reason you think you find order when you dig down far enough is that you systematically ignore any situation where you don't find order. Your categories have order built into them. You are drawn to order. There are probably a host of biases influencing that: availability, ontology, instrumentalism, and hindsight among them.
There's lots of order to be found. There is also infinite amounts of disorder, unprovable order, and alternate plausible order. Occam's razor helps sort it out-- that's also a heuristic.
Thanks, Eliezer. Helpful post.
I have personally witnessed a room of people nod their heads in agreement with a definition of a particular term in software testing. Then when we discussed examples of that term in action, we discovered that many of us having agreed with the words in the definition, had a very different interpretation of those words. To my great discouragement, I learned that agreeing on a sign is not the same as agreeing on the interpretant or the object. (sign, object, and interpretant are the three parts of Peirce's semiotic triangle)
In the case of 2+2=4, I think I know what that means, but when Euclid, Euler, or Laplace thought of 2+2=4, were they thinking the same thing I am? Maybe they were, but I'm not confident of that. And when someday a artificial intelligence ponders 2+2=4, will it be thinking what I'm thinking?
I feel 100% positive that 2+2=4 is true, and 100% positive that I don't entirely know what I mean by "2+2=4". I am also not entirely sure what other people mean by it. Maybe they mean "any two objects, combined with two objects, always results in four objects", which is obviously not true.
In thinking about certainty, it helps me to consider the history of the number zero. That something so obvious could be unknown (or unrecognized as important) for so long is sobering. The Greeks would also have sworn that the square root of negative one has no meaning and certainly no use in mathematics. 100% certain! The Pythagoreans would have sworn it just before stoning you to death for math heresy.
I wonder what your life must be like. The way you write, it sounds as if you spend a lot of your time trying to convince crazy people (by which I mean most of humanity, of course) to be less crazy and more rational, like us. Why not just ignore them?
Then I looked at your Wikipedia entry and noticed how young you are. Ah! When I was your age, I was also trying to convert everybody. My endless arguments about software development methods, circa 1994, are still in Google's Usenet archive. So, who am I to talk?
(Note: Mostly I write comments that complain about something you say, but please understand that there's a selection bias here. Even though I often find myself thinking "What an interesting way to think about that. Great idea, Eliezer!" I would rather write comments that have some kind of content, and those tend to be the critical ones.)
I don't understand why you invoke probability theory in a situation where it has no rhetorical value. Your conversation was a rhetorical situation, not a math problem, so you have to evaluate it and calibrate your speech acts accordingly-- or else you get nowhere, which is exactly what happened.
Your argument to your friend was exactly like someone justifying something about their own religion by citing their bible. It works great for people in your own community who already accept your premises. To anyone outside your community, you might as well be singing a tuneless hymn.
Besides that, the refuge available to anyone even within your community is to challenge the way that you have modeled the probability problem. If we change the model, the probabilities are dramatically changed. This is the lesson we get from Lord Kelvin's miscalculation of the age of the Sun, for instance. Arnold Sommerfeld once remarked that the hydrogen atom appeared to be more complex than a grand piano. In a way it is, but not so much once quantum mechanics was better understood. The story of the Periodic Table of Elements is also a story of trying different models.
Mathematics is powerful and pure. Your only little problem is demonstrating-- in terms your audience will value-- that your mathematics actually represents the part of the world you claim it represents. That's why you can't impose closure on everyone else using a rational argument; and why you may need a few other rhetorical tools.
Your confidence in your arguments seems to come from a coherence theory of truth: when facts align in beautiful and consistent ways, that coherence creates a powerful incentive to accept the whole pattern. Annoyingly, there turn out to be many ways to find or create coherence by blurring a detail here, or making an assumption there, or disqualifying evidence. For instance, you consistently disqualify evidence from spiritual intuition, don't you? Me, too. How can we be sure we should be doing that?
Why not learn to live with that? Why not give up the quest for universal closure, and settle for local closure? That's Pyrhhonian skepticism.
I think the advocates of Naturalistic Inquiry (see Lincoln and Guba) would say that you aren't talking about all of science, but of just the "positivistic paradigm" of science, whereas there is another paradigm called "naturalistic" or "constructivist" that does science differently.
I don't buy the whole Naturalistic program, but they raise some useful points. One of them is that the experiments you suggest require you to impose upon the object of your study an ontology along with the value system associated with it. When studying complex and ill-defined systems, such as psychological or social systems, this may suppress or disrupt the very phenomena that matter, and we never the wiser.
A naturalistic approach to science may tactically employ the kind of experiments you suggest, but proceeds with a great deal of caution about potential variables and hypotheses. "Hunter" and "gatherer" are socially overloaded terms with many implications and connections to other aspects of human life. It may be a big spaghetti mess to disentangle the issues. Inquiry proceeds in an exploratory fashion to tease out potential factors.
On other hand, it may not be a big mess! But the naturalistic bias is toward assuming complexity and subtlety and looking closely at the role of a priori assumptions in the choices of words, variables, and instrumentation that may lead to false results. It's sort of post-modernism applied to scientific method.
Again, I'm not a partisan of Naturalistic Inquiry. I just find it intriguing, and I have an allergic reaction to oversimplification (having been fooled so often by my own simplifications).