Q: How many theorems are there?

Posted on November 23, 2012 by The Physicist

Physicist: Every sub-field in math and physics has at least hundreds, and there are hundreds or thousands of sub-fields.  So overall we’ve proven… At least millions?

This is one of those things that can’t really have an exact answer, or even a ball-park answer.  One person’s theorem is another person’s corollary (“corollary” = “by the way”, in mathspeak).  More than that, not all theorems are winners.  For example, you’d be hard-pressed to find a professional nerd who thinks that the Pythagorean theorem isn’t a theorem, but something like “If N is an integer, then N(N+1) is even” generally isn’t considered profound enough for theorem status.  Not all theorems can be the BEST theorem.

So, counting theorems is kinda like counting art.  If you include finger-paint portraits and urinals, then the number of art gets a lot higher.  There have been lists of theorems made, but they’re always woefully incomplete.  Get a list, crack open any fairly specific/advanced math/physics book, pick a theorem at random, and it probably won’t be on the list.  At the very least, you can say that there are so many recognized theorems out there that no one could possibly live long enough to learn them all, or even any more than a small fraction.  You’d have better luck collecting all of the art.

Is it a theorem just because you hang it on a wall?

It’s been known for thousands of years that there are buckets of theorems, but it was generally assumed that at some point we’d find all of them and be done with math.  However, in 1931 Gödel added a pair of new theorems to the list, that was essentially about the list itself.  “Gödel’s incompleteness theorems” say, among other things, that there are an infinite number of axioms (“true, but unprovable statements”).  Extending that idea, we can expect that there should be an infinite number of theorems that relate these axioms to each other and talk about their consequences.  Math, and every strongly math-dependent discipline (which is all the good ones) are infinite, incompletable sciences.

In one wrenching insight, Gödel managed to simultaneously secure jobs for mathematicians in perpetuity and also to make their ultimate goal of proving/disproving everything utterly unattainable.

Posted in -- By the Physicist, Math, Philosophical | 6 Comments

Q: How much of a direct effect do planets and stars have on us? Is astrology reasonable or plausable?

Posted on November 18, 2012 by The Physicist

Physicist: Of the four forces: gravity, electromagnetism, nuclear-strong, and nuclear-weak, only the first two, gravity and EM, affect things over distances (at least, over distances larger than an atomic nucleus).  So, if the planets and stars have any direct influence on us it should be by way of one or both of those forces.

Clearly, it’s a good time to be a Leo.  Sucks for you, Aquarius.

The Moon’s gravity famously causes the tides, but the Moon’s electric field is effectively zero (but very interestingly, not exactly zero), and its magnetic field is random, scattered, and nearly non-existent.

The effect of gravity is much, much stronger than the effect of magnetic fields, and even the effect of gravity between planets is tiny. The bulk of those tiny forces is off-set by the fact that the Earth is free to move and fall through space (during free-fall is the only time you don’t feel gravity). This leaves only the much weaker secondary “tidal effects“, so called because they’re responsible for the tides which, although seemingly impressive, require very little force (specifically, the difference in the strength of the Earth’s gravity over the half-dozen feet of the tides).

The Moon’s gravity causes tides, but effectively nothing else. The Sun’s gravity has about 40% of the Moon’s influence and Jupiter, which completely dwarfs the effects of all of the other planets combined, has about 1 two-hundred-thousandth of the Moon’s tidal effect.

More than that, the gravity and electromagnetic influence of planets isn’t terribly surgical. Either will just pull on you in a very uniform way. They don’t grab a few cells at a time and re-write you love life or change your mood. Probably. Point is, our understanding of the known forces of the universe preclude the idea of the planets and stars having any direct influence on people.

Now to be fair, not being able to fit something into the current model doesn’t immediately exclude it from the realm of possibility.  For example, way back in the day the science fluid dynamics was really good at explaining how things like air and water move, but also “proved” that nothing should be able to fly.  Another beautiful example was a conundrum faced by geologists around 1900.  They had buckets of evidence that the Earth was at least hundreds of millions of years old (the ones who turned out to be right thought that the Earth was substantially older), and yet a back-of-the-envelope calculation showed that during that time the interior of the Earth should have cooled so much that volcanoes and other geothermal nonsense should be absolutely impossible.

But of course a quick look around shows a world full of birds and volcanoes.  So if there’s something around (birds and volcanoes), it really makes no difference what the prevailing scientific theory says one way or the other. Because reality wins.  It’s even written into the science charter, line one: “reality wins”.
By the way, the “flying issue” was later solved by taking into account rotational flows and viscosity, and the “warm Earth problem” is resolved by taking into account radioactive decay (which hadn’t been discovered yet).

So the better question isn’t “can so-called ‘science’ explain astrological effects?” but instead “are there astrological effects?”. There has been a lot of research into astrological phenomena, but so far all of the results have been negative or unrepeatable (science talk for “this isn’t a thing”).  Since the 18th or 19th century the scientific community has pretty much stopped looking, but they were at it for a very long time. There aren’t many scientific papers that seriously investigate this sort of thing, partly because the results are well-known, and partly because the experiments involved are easy enough that they tend to show up in middle-school science fairs relatively often (this is also why there are no articles in Nature about baking soda and vinegar volcanoes).

For example, just take all of the astrological predictions out of a newspaper (pardon: website) and read them in a random order to someone else (this is called a “blind experiment”) and see which one is closest to being accurate. You’ll find that the “correct one” is selected about once out of every 12 trials.

Long story short; whatever affect other planets and stars may have on us individually is completely drowned out by local “noise” (like the gravity and EM field of a passing truck), and worse there doesn’t seem to be an effect that needs explaining.

Posted in -- By the Physicist, Astronomy, Experiments, Physics, Skepticism | 59 Comments

Q: Why are scientists looking for life in space by looking for water? How can they be sure that all life uses water?

Posted on November 13, 2012 by The Physicist

The original question was:  I’m reading more and more lately about the findings of the Kepler satellite and that some scientists are estimating that roughly 5% of newly discovered planets in our galaxy may have mass similar to Earth, which always leads to the second question about how far they are from their stars and whether or not they might have liquid water.

Why are we so sure that life requires (liquid) water?  Seems a little Earth-centric to me (ie all life we know needs water so all other life must too).  Do they also need iphones?  Is there a good explanation for this grounded somewhere in physics?

Physicist: This is the central question of astrobiology.

There are two big reasons why space folk are looking for water.  The first and obvious one is that, without exception, ever form of life we’ve ever found has required water.  Admittedly all of those forms of life were found here on Earth, but it still holds true.

In an attempt to combat this, biologists have been scouring the planet looking for “shadow biospheres” and “extremophiles“.  For example, it was once believed that nothing could survive in extreme radiation, or boiling temperatures, or without sunlight or oxygen, but each of these have been found to be false.

Those red things are Giant Tube Worms, animals living on the hydrogen sulfide streaming out of this black smoker on the bottom of the ocean.  It’s very dark, very hot, very poisonous, crushingly “pressurous”,  and even fairly radioactive near these vents.  But the Giant Tubes Worms ain’t shook.

For example, life (sometimes even complex life!) has been found thriving deep underground, in anoxic basins (“puddles” of dense, ultra salty, oxygen-free water scattered on the bottom of the ocean), and famously around and even inside of black smokers on the bottom of the ocean.

Life is pretty sneaky (and gooey, more often than not).  Recently life was found in Mono Lake that can use arsenic instead of phosphorus when it has to.  There are even bacteria that have found clever ways to live inside solid ice (Technically those clever bacteria melt the ice).

Excursions have been made to places like Death Valley or the Atacama Desert to find life that doesn’t use water.  Both places are extremely dry, but otherwise aren’t so bad.  Unfortunately, so far everything has come back negative.  Living things can survive pretty much anything except for being thirsty.  So that seems to be the line in the sand (mud).

The second reason that astronomers are looking for life by looking for water is that you can’t look for something if you don’t know anything about it.  If someone asks you to go into a room and “find the keys”, you can do it.  But if someone sends you into a junk-shop and asks you to “find that thing”, you’d have no way of knowing if you’ve successfully found it or not.

“Good day to you shopkeep!  I’m looking for a thing, but I don’t know what it is.  So, where is it?”

If there’s life out there that doesn’t involve water, we have no idea what it’s like.  Maybe giant crystal formations that reproduce on geological time scales?  Maybe parts of the weather system are living organisms (somehow)?  Maybe there’s life similar to us, but that uses something like liquid ammonia instead of water?  Giant transforming robots?  Who knows.

Worse than that, we have a hard time even defining what life is.  It’s surprisingly difficult to come up with a definition for “living” that includes things like viruses and dormant spores, but that doesn’t include fire.

So, it’s not that we’re sure that life requires liquid water, we just don’t know what else to look for, and liquid water and life seem to always go hand in hand (here on Earth at least).

The junk shop photo is from here.

Posted in -- By the Physicist, Astronomy, Biology, Logic | 7 Comments

Q: If energy is neither created nor destroyed, what happens to the energy within our bodies and brains when we die?

Posted on November 8, 2012 by The Physicist

Quick note: If you’re presently grieving, don’t read this.

The original question was: If energy is neither created nor destroyed, what happens to the energy within our bodies and brains when we die?  I think I understand that the metabolic energy tied up in our cells will be used in the decomposition process, but what about the electrical energy in our brains/bodies?  This would seem to be a measurable amount of energy that at the moment of death is no longer required by the body/brain and would have to go somewhere.  I’m not asking from a theological or spiritual perspective, but strictly as a question of physics.

… [is there] a measurable radiation of heat at the moment of death.  Do you know if there have been experiments that have measured the heat loss and correlated it to the known amount of electrical energy in the human body?

Physicist: Electrical energy is nothing special.  Just like the chemical energy in our bodies, it breaks down into heat.  For example, the heat given off by light bulbs (or electric heaters for that matter!) is a result of electrical energy.  When electricity is flowing to a light bulb, that’s where the electrical energy is going; it’s turning into light.  When you pull the plug (so to speak) what tiny, tiny amount of electrical energy there is in the wires runs out almost immediately.

The term “electrical energy” is actually a little vague.  So, to be specific, in our nervous system there are tiny ion pumps that maintain an imbalance of charges between the inside and outside of the nerve cells.  When a nerve cell fires, charges are allowed to suddenly flow through the cell membrane in a process called an “action potential“.  The way electricity flows along nerve cells is different from the way it flows down a telegraph wire (“inside-to-outside” instead of “along”), but whatever.  The point is, there are mechanisms that maintain an imbalance of charge (which is electricity waiting to happen), and that imbalance is drained a little bit every time the nerve fires.

Death (excluding spectacular deaths) isn’t instantaneous.  In fact, what with medical science, it’s become more and more difficult to even define when people are dead.  Time was you could define death as being a lack of heart beat, but people have come back from worse (by that metric, Dick Cheney has been dead for a while).  Death is more of a break-down of the whole system, as opposed to a sudden event.  The heart stops doing whatever hearts do when they’re not loving, oxygen and nutrients stop going where they’re needed, and in short order the nerve cells in the body lose the wherewithal to pump ions.  Like batteries that are no longer being recharged, they run down.  Nothing special.  Like every kind of energy, whether electrical, kinetic, sonic, or sports fever, the electrical potential in the body eventually becomes heat energy (it’s an entropy thing).

The energy we “carry around” takes the form of chemical energy like fats and sugars.  When our nervous system creates electrical energy we lose an equal amount of chemical energy.  So, rather than being energy itself, life is all about moving energy around from one form to another.

What this question is clearly really about is the fact that it seems as though there’s a fundamental difference between animate and inanimate people.  Admittedly, dead folk are a hair less energetic than living people (with some exceptions).  There are a few kinds of energy (surprisingly few), but spiritual energy doesn’t seem to be one of them.  In terms of physical energy, the difference between a living body and a very recently dead body is just a question of how that energy is being organized.  Living critters in general are very good at using chemical energy for things like moving, growing, etc.  Newly dead critters have about the same amount of chemical energy, it’s just that they don’t use it.  Instead, whatever comes along to consume the body uses it (whether that’s fire or decomposition or whatever).

There have been many, surprisingly callous, attempts to measure a drop in energy and/or mass leaving the body at the so-called “moment of death”.  However, these experiments have been vague and, much worse, unrepeatable.  The most famous is the experiment by Dr. Duncan MacDougall in which, by putting patients dying of tuberculosis on giant scales, he found that those patients lost 21 grams on average between life and death.  To be fair, homeboy had 6 data points (that is: people) and a lot of statistical noise, so his conclusions have about the same amount of statistical weight as “vaccines cause Autism“.  To date, there are no confirmable experiments that show that anything special happens during death, other than a general “shutting down”.  In particular, nothing that’s both “inspiring” and verifiable seems to suddenly leave the body when we die, materially or energetically.

Posted in -- By the Physicist, Biology, Physics, Skepticism | 276 Comments

Q: Could Kurt Vonnegut’s “Ice-9 catastrophe” happen?

Posted on November 3, 2012 by The Physicist

The original question was: Is it possible to actually create a substance such as Kurt Vonnegut’s “Ice-9 (Nine)“, which could, in theory, bond with water (especially seawater) and replicate like a virus, freezing the oceans (or all liquid water on Earth) solid?

Physicist: Nope.  There actually is an ice-IX, but there’s nothing that special about it.

Crystals in general are self-propagating.  That is, once a crystal is present the raw materials in the environment find that falling into place in the crystal’s lattice results in a net drop in energy, but weirdly enough it usually takes a little energy to start a crystal from scratch.  You can see this in action with super-cooled water: the water is cold enough that it should freeze, but no collection of water molecules is able to start the ice crystal.

Super-cooled water freezing on contact with previously formed ice.

The first tiny crystal that gets the ball rolling is called a “seed crystal”.  Often it’s unnecessary because when a bunch of atoms are ready to crystallize small flaws in the environment can “look” like pieces of crystal to them, and they’ll just grab on to them (these are called “nucleation sites“).

It may seem strange to talk about a different kind of water-ice other than just regular “ice”, but keep in mind that many substances are perfectly happy to crystallize in multiple ways.  For example, sapphires and rubies are made of exactly the same stuff, aluminum oxide (“aluminium” for our not-American readers), just put together in different ways due to the different environments that they form in.

Aluminum oxide in sapphire and ruby forms.

Water in particular crystallizes in a surprising number of ways.  To date, 15 different forms of water ice have been found, however here on Earth only one kind (“ice-I”) can form naturally.  Even at the bottom of the Mariana Trench the pressure isn’t nearly high enough for any other kind of ice.

The different forms of water and ice at different pressures and temperatures. Sea-level air pressure is the horizontal red line.

A general rule of thumb is, if something wants to crystallize, and it’s in an environment where it can do so, eventually it does.  Even if there’s no seed crystal already present, at some point a nucleation point is bound to show up (given enough time) and a crystal will form and grow.

The whole idea of Vonnegut’s ice-9 is that it’s a crystalline form of water, that grows rapidly at reasonable, Earth-like pressures and temperatures.  If it could form at all, then at some time in the last several billion years, at some place in our gargantuan oceans, a tiny ice-9 crystal should have formed accidentally and encompassed the world.

We’ve already seen this on other, aptly named, ice-worlds.  For example, Europa has lots of water, and is in an environment (pressure and temperature) conducive to the formation of ordinary ice-I.  Not surprisingly, Europa long ago underwent a colossal “ice-9 type catastrophe”, in that all of its water (at least on the surface) quietly froze solid.

Long story short, if the Earth’s water could somehow be induced to crystallize at room-temperature, it almost certainly would have done so at its earliest convenience, a very long time ago.

The very cool super-cooled water gif is from here.

Posted in -- By the Physicist, Paranoia, Physics | 15 Comments

Q: How accurately do we need to know π? Is there a reason to know it out to billions of digits?

Posted on October 29, 2012 by The Physicist

Physicist: For essentially every imaginable purpose, knowing that π ≈ 3.14159 is more than good enough.  After all, every additional digit you have yields ten times the accuracy.  So if you know π out to twenty digits, that’s not 20 times more accurate than just “3”, that’s 100,000,000,000,000,000,000 times more accurate.  Every major civilization has been aware of π and have relied on various approximate values.  Basically they’d find an approximation that was “good enough” to find the length around any circle given the length across it (the definition of π) and go with that.  For example, in parts of ancient India they used π ≈ 62832/20000 (5 digit accuracy), for a while in ancient Egypt they used π ≈ 22/7 (3 digits), and π = 3.2 (1 digit) was nearly used in Indiana embarrassingly recently.

π has a true, fixed value, but in practice it only needs to be approximated.

When you’re figuring out how much rope is needed to wrap around a barrel (or whatever they needed π for a few thousand years ago), knowing it out to 10 digits or more is massive over-kill.  With only 32 digits of π (3.14159265358979323846264338327950), and a really good measurement of the diameter of the Milky Way galaxy, you could wrap a rope around the galaxy’s circumference that’s accurate to one atom in length.  That’s too accurate.

So how do we know that π is irrational?  And why does that matter?  It wasn’t until 1761 that π was proven to be irrational, so those previous civilizations can be forgiven for not knowing that any attempt to explicitly write π will fail (when written, π meanders on forever without any particular pattern, so no attempt to explicitly write it will ever work).  There are several proofs of π’s irrationality but they’re all fairly meandering and complicated (even for this website), although if you’re comfortable with calculus, running through them is a good exercise.  In practice (when building physical things) it couldn’t matter less that π is irrational.  However mathematicians get very excited about these sorts of problems.  In fact, it can be shown (without too much effort) that \pi = 4-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\frac{4}{11}\cdots, and because of this a relationship between the infinitude of primes, and the complicated nature of π can be shown (that is, if there were a finite number of primes, then π would be a rational number).  More recently it was shown that π cannot be the solution of any polynomial (with rational-numbered coefficients), and there’s been a lot of work on whether or not any random sequence of numbers shows up somewhere in π (if you’re willing to look long enough).

To date, what with computers and clever mathematicians, π has been calculated out to several trillion digits (based entirely on its extremely simple definition, and buckets of math).  The summation series above is one of the easiest to remember, but it’s very slow.  There are others that approach π much, much faster and can be found here (and are responsible for our ability to calculate π with such stupefying accuracy).  But ultimately there are two reasons to know π out to more than a handful of digits: studying π for purely mathematical pursuits, and winning competitions.

Funny because it’s true.

The webcomic is from here.

Answer gravy: π = 3.14159265358979323846264338327950288419716939937510582097494459230781
6406286208998628034825342117067982148086513282306647093844609550582231
7253594081284811174502841027019385211055596446229489549303819644288109
7566593344612847564823378678316527120190914564856692346034861045432664
8213393607260249141273724587006606315588174881520920962829254091715364
3678925903600113305305488204665213841469519415116094330572703657595919
5309218611738193261179310511854807446237996274956735188575272489122793
8183011949129833673362440656643086021394946395224737190702179860943702
7705392171762931767523846748184676694051320005681271452635608277857713
4275778960917363717872146844090122495343014654958537105079227968925892
354201995611212902196086403441815981362977477130996051870721134999999… and so on.

Update: Actually, the proof that pi is irrational isn’t quite too meandering and complicated.  One of the simplest is now a post!

Posted in -- By the Physicist, Math | 12 Comments