Q: What is the meaning of life?

Posted on October 29, 2009 by The Mathematician

Mathematician: I’m glad you asked. The theory of evolution with natural selection sheds some light on the question of why humans exist, which in turn relates to the meaning of life. First of all, let me get this out of the way: while evolution is still called a “theory”, it has a tremendous amount of evidence in support of it (including gradual transitions in the fossil record, radiocarbon dating, DNA analysis, laboratory experiments, etc.) and, as biologist Richard Dawkins is known to say, it is only a theory in the sense that the “theory of gravity” is a theory. From a scientists perspective, evolution is a fact. But what does that have to do with the meaning of life? Well, evolution tells us that human beings share a common ancestor with apes, not to mention with pigs, dogs, cats, rats, plants, and bacteria. If historically the conditions on earth had been very slightly different than they were, the best traits for survival would have been different also, and therefore we would expect that some other species besides humans (possibly with intelligence as great as ours) would now dominate this planet. Hence, evolution tells us that humans have the capacities that they do now simply because those capacities helped our ancestors survive long enough to have children, or made them more effective at finding mates.

I believe most people will agree that “What is the meaning of life?” is a question that is meaningless when it is applied to the lives of rodents, insects, or bacteria. As we are simply evolutionary offshoots of these creatures, what makes us think that this question will have any more significance when applied to us? The primary characteristic that differentiates us from these other creatures is our powerful brains that have incredible capacity for abstract though (including the ability to consider questions like “what is the meaning of life?”). Each of us exists today because our ancestors managed to survive. Their survival occurred both because they were well adapted to their environment, and because they got very, very lucky. Our existence then is, in some sense, a happy accident, and lacks the deep cosmic significance that questions like “what is the meaning of life?” presuppose. Fortunately, however, our brains are very adept at discovering meaning in all sorts of places. Life does not require an all encompassing, universal meaning or purpose in order for us to find that our own lives are meaningful, and that is a truly wonderful thing! We can feel totally fulfilled despite being little more than happy accidents of evolution. Of course, “meaning” is a very real and important emotion, but that does not make it an objective property of things. Fortunately, there is no reason to fret over life itself having no ultimate purpose. We ourselves can find something that fills us with a sense of purpose, which for practical purposes is just as good.

Physicist: Every now and again a question comes along that implies more than it asks.  Questions like: “What’s that blue thing?”, or “Who ate this?”.  If there’s nothing blue or eaten around, then these questions don’t make sense.  The same is true of the classic: “What is the meaning of life?”.

Posted in -- By the Mathematician, -- By the Physicist, Evolution, Philosophical | 17 Comments

Q: Why is e to the i pi equal to -1?

Posted on October 29, 2009 by The Physicist

Physicist: This equation (e^{i \pi} = -1) was recently voted one of the most famous equations ever.  That isn’t part of the answer, it’s just interesting.

First, you’ll find (by plugging them into a graphing calculator and graphing) that:

1) Sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} - \cdots

2) Cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \cdots

3) e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots

Where N! = 1 \cdot 2 \cdot 3 \cdots N.

These are called “Taylor expansions” of “Sine”, “Cosine”, and “e to the x”.  If you were to continue the patterns above forever, then you would find that the equality is exact.  There is some very exciting math to back me up on this, but for now just trust.

Know that i^2 = - 1.  This is how you define i.  Therefore, i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1, i^5 = i , \ldots

Check it!:

e^{i x} = 1 + (ix) + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \frac{(ix)^5}{5!} + \frac{(ix)^6}{6!} + \frac{(ix)^7}{7!} + \cdots = 1 + i x + \frac{i^2 x^2}{2!} + \frac{i^3 x^3}{3!} + \frac{i^4 x^4}{4!} + \frac{i^5 x^5}{5!} + \frac{i^6 x^6}{6!} + \frac{i^7 x^7}{7!} + \cdots = 1 + i x - \frac{x^2}{2!} - i \frac{x^3}{3!} + \frac{x^4}{4!} + i \frac{x^5}{5!} - \frac{x^6}{6!} - i \frac{x^7}{7!} + \cdots

and grouping terms that have “i” in them:

= ( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots) + i ( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots )

= Cos(x) + i Sin(x)

Holy crap! e^{ix} = Cos(x) + i Sin(x)!

This is the “Euler Equation”.  Or one of them at least.  Just plug in “x = \pi“.

e^{i \pi} = Cos(\pi) + i Sin(\pi) = (-1) + i(0) = -1.

So the trick is Euler’s equation, which is (surprisingly) true.

Posted in -- By the Physicist, Equations | 30 Comments

Q: How does a refrigerator work?

Posted on October 29, 2009 by The Physicist

Physicist: “Making cold” is impossible, so a “refrigerator” is really just a heat pump.  Scientists don’t talk about much beyond of a handful of thought experiments.  Quantum physicists: double slit, relativists: trains, thermodynamicists: mirrored boxes and pistons, etc..  Refrigerators can be explained almost entirely by pistons.

When a piston is compressed to half it’s original volume, the temperature of the gas inside the piston doubles.  This can be viewed as either “concentrating the energy” of the gas, or, if you’re actually doing the math, you can look at the energy the piston imparts on the gas by moving inward.  Essentially, the gas gains energy from the moving piston the same way that a tennis ball gains energy from a moving racket or a baseball gains energy from a moving bat.  Conversely, if a gas is allowed to expand, it will cool.

The basic heat pump. This same basic structure is also found in AC units, car radiators, and just about every other thing that makes coldness.

The basic heat pump. This same basic structure is also found in AC units, car radiators, and just about every other thing that makes coldness.

1) A (very) cold gas moves through the tubes in the back of the freezer, absorbing heat.  The tubes absorb heat because, as cold as the freezer is, the the tubes are colder.

2) The (now slightly warmer) gas runs into the compressor (which compresses, and is the thing that makes that humming sound, and in the picture above is labeled “pump”).  Compressing the gas heats it up.  The gas then passes into the radiator coils (which radiate, and are found on the back).

3) Once the gas loses heat to the surrounding air it drops to near room temperature.

4) The compressed, room-temperature gas now passes through an expansion valve (which is a fancy word for “spray nozzle”).  Expanding causes the gas to cool (a lot), and it is now ready to absorb heat from the freezer.

5) Goto 1.

Posted in -- By the Physicist, Engineering | 4 Comments

Q: How do I find the love of my life?

Posted on October 25, 2009 by The Physicist

Physicist: If you assume that there’s one person “out there” for you, and you share a deep connection, then go with that.

If you don’t believe in the “deep connection” part, then you’re shit out of luck.

The question I can answer is: “Of N (ladies and/or gentlemen, hereafter “peeps”), how do I find the best (for me)?”.  This question is now essentially the “Secretary Problem“, which happily has an ideal solution.

1) Get N Peeps.

2) Date N/e of them (e=2.718281…).

3) Continuing dating, but the first Peep you meet who’s better than all of the first N/e Peeps, you marry.

4) Stop dating.  This is arguably the most important step.

This procedure has a success rate of about 37% (1/e).  Good luck out there cats and kittens!

Posted in -- By the Physicist, Philosophical | 10 Comments

Q: Why does “curved space-time” cause gravity?

Posted on October 25, 2009 by The Physicist

Physicist: In a flat space local ideas about “parallel” and “perpendicular” are global.  That is, if two lines are parallel, and you follow them for a while, then they’ll still be parallel.  (By “flat” here I mean exactly this property, parallel is parallel forever.  Not just “flat like paper”.  So you can have 2-d flat space, 3-d, 4-d, whatevs)

An example of curved 2-d space is the surface of a ball (just the surface, don’t worry about the inside and outside).  If you draw two parallel lines on the ball, then eventually they will cross.  The curvature forces the two lines to come together.

Not all roll-over text is worth reading.

A straight line on a sphere always traces out a "great circle", like the equator. These lines are parallel twice, but also intersect twice.

If an object is not experiencing any force, then it will travel in a straight line through space.  This is true for space-time as well.  So if you’re sitting still (traveling forward in time), and no one applies a force to you, you’ll continue to sit still (travel forward in time).

Now imagine two people hovering above opposite sides of the Earth.  I say hovering in place because this means the lines they trace out in space-time are (initially) parallel.  As you run time forward you’ll notice that, even though no force is acting on them (don’t say gravity) and they are traveling in straight lines through space-time, they still move together (fall toward the Earth).

This is due entirely to the Earth curving space-time around it.  Literally, it takes the original “flatness” of empty space, and curves it.  It’s a little more complicated because the time dimension and the spacial dimensions are fundamentally different, but not as much as you’d think.

Another, slicker-sounding way to describe gravity is: “Things fall because time points a little bit down”.  That’s not creative prose, I mean that literally.

Posted in -- By the Physicist, Relativity | 31 Comments

Q: What is monotony?

Posted on October 21, 2009 by The Physicist

Physicist: It seems fair to say that monotony goes hand in hand with predictability which goes hand in hand with low entropy.  So (mathematically speaking), you can reasonably define monotony as the reciprocal of entropy, or something like that.

Posted in -- By the Physicist, Entropy/Information, Philosophical | 4 Comments