I get frustrated when people say that a certain bit of information is “unknowable” when they mean “unknown…for now”, but here is a list of 10 things the author claims is totally unknowable, and he seems to make a good case for each…at least to this layman.

Are each of the ten truly unknowable(the answers can never be known no matter who much time and effort is put into their study)?

Are there other things that are unknowable

Can you know what your tongue tastes like? Can you know what your nose smells like?

“On the subject of stars, all investigations which are not ultimately reducible to simple visual observations are … necessarily denied to us. While we can conceive of the possibility of determining their shapes, their sizes, and their motions, we shall never be able by any means to study their chemical composition or their mineralogical structure … Our knowledge concerning their gaseous envelopes is necessarily limited to their existence, size … and refractive power, we shall not at all be able to determine their chemical composition or even their density… I regard any notion concerning the true mean temperature of the various stars as forever denied to us.”

Auguste Comte

I think this is more along the lines of “Unknowable by anybody, not just by certain individuals”-another person can find out the answers to those questions…or you could snip off a bit of your tongue and have a taste, or seal off every part of your face except your nose and have a quick sniff when close to a reflective surface.

Is in inconceivable that a method to visually observe the stars will someday be possible? I would stick that one in the “Unknown…so far” category.

edited to add: The man lived in the first half of the 19th Century-I wonder what he would say today about possibilities?

Obviously he was very wrong about us not being able to tell anything about the chemical composition of stars; spectroscopic analysis of light was being developed while Comte was alive, I think.

One thing about the points made in the OP. Yes, there are computer programs whose halting problem is undecidable and those programs can never halt, obviously. But by that very fact, it is impossible to look at a program and say, "Ah the halting problem for *this program* is undecidable. Roger Penrose fell flat on his face by assuming there were (in *The Emperor’s New Clothes*, thus assuming that human’s could compute something computer’s cannot.

I’ll give it a shot.

Saying we don’t know Graham’s number seems silly, since we can write out a formula to produce it. It’s like saying we can’t know pi. Sure, not exactly, but we can approximate it.

I also don’t really feel that paradoxes are unknowable. It’s not that we don’t know them. It’s that they contradict themselves. There can be no set of sets that don’t contain themselves, and there can be no smallest number not expressible in eleven words or fewer. It’s weird saying they are unknowable because they don’t exist.

We can know if a computer program will stop–by running it. That one is just incorrectly worded. We can’t know if a program can stop without running it.

I guess you can argue that that there are an infinite number of numbers that we can’t compute, giving you an infinite number of unknowable things, which they just lumped into one for convenience. But the explanation is wanting, and actually describes some infinities of the same size, e.g. the number of even numbers and the number of integers. (Proof: for every possible integer, you can multiply it by 2 to get an even number. Hence there is a 1 to 1 mapping.) You actually get into different concepts of numbers (like density) to actually understand the differences.

Not sure on the next one, so I had to go read up on it. I’ve known about Godel’s incompleteness theorem, but am not sure about the explanation. From what I read, if we consider “mathematics” to be a system of axioms, and not a set of different systems, then it works. But the thing is that we can change the axioms. And, of course, none of his examples are known to be unprovable, but I assume that was intentional: he was just giving possible examples.

Tarski’s proof is just a repeat of the above, in my opinion. Arithmetic is a set of axioms, and can’t prove itself. It needs a set of axioms above itself–a meta-language.

I will give him Heisenberg’s Uncertainty principle, I guess. More exactly, there is a limit to the precision we can determine, and it spread across both momentum and position. And I could see an argument that this is just like “not knowing pi” as above, but, for some reason, this seems more like actually unknowable, since we can actually get more precision in one if we give it up for the other.

Chaitan’s constant is just a repeat of the halting problem, in the same way Tarski is just repeating Godel. It shouldn’t count twice.

I guess “will be unknowable” counts. But I question whether this is just a limit to our own science now that we think this will happen.

So, looking back over all this, I count at most 4, plus an infinity of numbers we just can’t compute or otherwise represent.

Maybe I’m misunderstanding you, but it’s not true that all programs which are undecidable never halt. Undecidable means that we can’t know, even in theory, if they’ll ever halt.

What it means is that there is no algorithm that can decide, for all possible programs, whether or not that program will halt.

This is of course a theoretical thing that is true for a machine with unlimited physical resources. For any real world machine with finite physical resources, you can say that if it runs for at least X cycles then it will never halt (because its deterministic and all possible states will have been exhausted). However, this isn’t very useful because X cycles may be longer than the life of the universe.

This program I just wrote has been running for three days. Will it halt?

Yes. Eventually you’re going to have to stop it to upgrade the software on your computer.

Which completely misses the point.

I ran a program the other day, and it halted soon after. I’ve solved this allegedly unsolvable problem with a resounding yes.

I don’t care about your program.

What I am going to do tomorrow.

The point of **Andy L**’s quote from Comte was that Comte, using the knowledge of his time, blithely asserted that many things were unknowable. Those same things are now not only knowable, but well known.

The point is that it’s a lot harder to discern the difference between unknown yet and unknowable than it at first appears. Our best attempts to be objective about our knowledge and ignorance are still heavily conditioned by our current knowledge and current ignorance.

My take on the overall topic is that there’s a certain mysticism or romanticism associated with the idea of unknowable. Which makes looking for those things much more psychologically attractive than it is actually useful. It certainly works to sell pop-sci books and Discovery Channel episodes.

ISTM that it’s smarter to simply assert that our current knowledge has boundaries and we can push almost all of them outwards if we try. Minus some islands of unknowability that we’ll only recognize as actually unknowable in hindsight.

e.g. the “mystery” of the Uncertainty Principle is pretty well debunked. It’s a trivial consequence of some basic math on what’s actually a composite property of matter. Here’s a nice explanation within a thread that was itself starting from the mystical unknowable POV and was pretty well reduced to “it’s just math; basic math” by the 2nd post: http://boards.straightdope.com/sdmb/showthread.php?t=832204

Yes (and thanks - I should have made that clear, rather than just dropping in the quote, but I was in a hurry at the time).