[rvwadcot] Irrational numbers and dark matter

With fractions of greater and greater denominators, you can specify numbers on the real number line with greater and greater precision.  There seems to be no limit.  How weird it is to discover that the square root of 2 cannot be expressed as a fraction, no matter how great the denominator.  It's a number, you can measure it, but it's unlike any other rational number.  Then, we discover, much later, that "most" numbers are irrational numbers.

This kind of parallels the astronomical discovery of dark matter.  It obeys gravity like all other matter but is unlike any normal matter we encounter in our lives.  And then we discover most of the matter in the universe is dark matter.

After a while, we get a good handle on irrational numbers like pi, e, and sqrt(2).  Computers can calculate billions of digits of them as well as many other irrational numbers using algorithms such as Newton's method and summing Taylor series.

How weird it is discover that there exist irrational numbers that no computer can compute.  A famous one is the probability that a random Turing machine will halt (Chaitin Omega).  We'll define what it means to "compute" an infinitely long number in a roundabout way: to be able to recognize, after a finite amount of time, whether a given number is NOT the number we want to "compute".   It's OK if the computation runs forever if we give it as input the correct number; one can imagine it comparing digit by digit until a difference is found.  Complexity class is co-R.E. (the complement of recursively enumerable).

You can guess the punch line: "most" real numbers are not computable.  Most numbers are literally beyond imagination, defining "imagination" to mean a computation by any computer, including the biological computer that is the human brain.

Pick a number between 1 and 10.  There are 10 integers, lots of rationals between the integers, lots more computable numbers between the rationals.  But most of the "space" between 1 and 10 is filled with unimaginable monsters.

After a while, we got decent handle on dark matter.  Maybe they are MACHOs, but probably they are WIMPs, a yet-undiscovered subatomic particle which, like a neutrino, rarely interacts with normal matter.  In the Bullet Cluster, we actually got a picture of dark matter separated out from normal matter.

How weird it was to discover next dark energy, an energy that seems to repel space itself, making the universe expand faster and faster.  Dark energy is fundamentally mind-boggling.  It behaves unlike anything else known to science, seemingly disobeying the law of gravity and the conservation of mass-energy.  Like an unsatiable monster, it will eventually destroy everything. And like the punch line for uncomputable numbers, it turns out that most (70%) of the mass-energy of the universe is dark energy.