Minus Times Minus Equals Plus

Here’s an example I like of a negative number times another
negative number equaling a positive number:

I have been giving away five dollars each minute.
Currently, at time t = 0, I have zero dollars, but I am continuing
to give away five dollars each minute in the form of IOU’s.
Thus, the equation for the number of dollars I have at time t is
D = -5t, where D represents dollars.

Using that equation, calculate how many dollars I had four
minutes ago.


by Bob Day
Copyright (C) June, 2012 by Bob Day. All rights reserved.

Time is weird.  Maybe not as weird as consciousness, but definitely weird.  But there are some simple things we can observe about time.  First, it seems to be the property of our universe that provides the capacity for change. Without this capacity, things could not change.  We see a car moving.  "Now" it's at one spot.  A little bit "later" it has moved a little farther down the road.  Without time it couldn't do that — it would remain frozen in place.  Question: Does time always involve motion?  I think maybe it does.  Suppose we're listening to music.  The sound is changing.  The sound is a sensation in our heads, and apparently not moving.  But what causes the sound is atoms vibrating — moving back and forth — in the air.  So sound requires motion to make it.  I'm pretty sure it's all like that — same thing with a leaf changing colors: chemical reactions are involved, and, consequently, motion of atoms.  Maybe you can think of a counterexample — an example of some kind of change that doesn't involve the motion of something.

Here's another weird thing about time: It seems to go at the same rate for everything.  The definition of one second is this: "The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom (at 0 degrees K)."  OK, we have a stopwatch and we'll clock our cesium 133 atom.  We time one second on our watch and count.  Sure enough — 9,192,631,770 periods!  Let's do it again.  We get the same thing, 9,192,631,770 periods.  So time went at the same rate in each of our measurements.  But wait — that's our definition of time: whenever the cesium 133 atom makes 9,192,631,770 transitions, that's one second, no matter how long it takes.  So our first measurement might have taken a second, and the second measurement, 100 years.  Except that, if that were the case, our stopwatch (and everything else) would have had to slow down too.  That's what I meant when I said that time, apparently, goes at the same rate for everything.  And if that rate changes, it changes for everything at once.

So, as far as we can measure, time seems to flow at the same rate for everything, at least in our neck of the universe.  But who is to say that time might not flow at a different rate in a chunk of space (in our same relativistic frame) a zillion or so light years away from us?  The immediate consequence of that might be that the speed of light in that chunk of space would be different from our speed of light.  Possible?  I couldn't say either way.  But, as far as I know, there is no rule that says the flow rate of time has to be a constant throughout the universe.


Vitamins for the Mind

                               The 4:00 Meeting
Tom and Bill are standing at two places on a straight road.  Tom starts
walking toward Bill and arrives at Bill’s original place 11 minutes after
Bill had left.  At perhaps a different time, Bill starts walking toward Tom
and arrives at Tom’s original place 15 minutes after Tom had left.  When
each reaches the other’s original place, he immediately turns and starts
back, and they meet in the center at 4:00.  Assuming that they walked at
constant rates, when did each start to travel on the road?

                          Alice Forgets Her Purse
On her way out of Macys, Alice walks down a down-moving escalator
in 50 steps.  When she reaches the bottom, she suddenly remembers
she forgot her purse, and she turns and runs back up the escalator in
125 steps, stepping five times as fast as she went down.  How many steps
are on the surface of the escalator?

                               The Twelve Coins
You have 12 coins.  They are identical, except that one of them is either
heavier or lighter than the rest.  In three weighings on a balance scale, find
the odd coin and whether it’s heavier or lighter.

Consciousness Totally Explained and Elucidated

Consciousness Totally Explained and Elucidated

Copyright (C) April, 2012 by Bob Day.
All rights reserved.

Like I have even a clue.  I don't think anyone does.  But I do have some thoughts about it.  Here's the definition I'm using: "Consciousness" — that mysterious phenomenon by virtue of which we can say "I" in our minds.

1. First off, you've gotta wonder why consciousness is even necessary — why it's useful from the standpoint of evolution; that is, what survival benefit does it confer?  From a survival point of view, why wouldn't we do equally well just receiving and reacting to stimuli in a totally automatic sort of way? — like we assume a computer does.  It could be a pretty complex path between the stimulus and the reaction — for example, a computer can play an excellent game of chess, but it's still completely automatic and "mindless".  I have no clue on this point.  But "Nature" does seem to have gone to quite a bit of effort surrounding its implementation of consciousness — it's given us an elaborate internal "display screen", or in Microsoft Windows terminology, an internal "desktop", that allows us to speak with ourselves internally, to see images in our minds, and to "think".  So, obviously, there is an evolutionary benefit to incorporating consciousness into the design of creatures over a purely automatic, reactive design.  Or is there?  Anyway, my own suspicion is that consciousness is just a cheap solution to an evolutionary problem.

2. Questions without answers.  Suppose we took a person's brain out of their head, took the brain apart, cell by cell, neuron by neuron, put it back together again, and put it back into the person's head.  Would the person have consciousness?  Would he or she be the same person with the original consciousness (whatever that might mean) ?

Suppose we took a person's brain apart, and duplicated it cell for cell.  Then we put the duplicate brain back into the person's head.  Would the person have consciousness?  Would he or she be the same person?

Suppose we put the original brain into one body and the duplicate brain into another?

Suppose we took a persons brain apart, duplicated it cell for cell with semiconductor chips, and hooked up the semiconductor brain into the person's body?

3. Suggestion for an experiment.  I thought about the idea of two people having duplicate brains a little bit more, and I can imagine one result being that they would share a consciousness. (Note – I'm got going so far as to say that this is a real possibility — that's why I used the word 'imagine'.)  Going a little further, maybe creatures get consciousness somewhat analogously to how a radio receives a radio signal.  Maybe we are all tuned to some kind of central source of consciousness.  Maybe each of us is tuned to a different "frequency" of that central source, with the result that each of us has a separate consciousness.  So maybe if two people have exactly duplicate brains, they are both tuned to the same "frequency", and thus share a consciousness. (It's my blog, so I can take these flights of fancy!)

There's just a smidgeon of evidence that this is a possibility: Many sets of identical twins say that that they have a sense of one another.  One can sense when the other is in danger, can feel the other's emotions, can "feel the other's pain".  Perhaps the brains of identical twins are tuned very close to the same "frequency", so their consciousness overlaps to a degree.  As far as I know there is no good experimental evidence for this.  In fact, as of April 2012, James Randi's million dollar prize for reliably demonstrating any kind of paranormal phenomena has gone unclaimed for 14 years.  The evidence, such as it is, is all anecdotal.  That doesn't mean it's worthless, but it does mean we're not on solid ground.

My idea for an experiment is to nail down what I said in the previous paragraph, one way or the other.  Get some cloned pairs of mice that are as close to being exact duplicates as possible.  Then take one mouse of each pair, give it a little electric shock, and see whether there is any reaction from the other one in the pair.  Or, teach one mouse in each pair to run a maze, and then see whether the other mouse in the pair can learn the maze faster.  Of course, we have to have all the usual experimental controls: mice in each pair separated; a control group of mice, etc, etc.


Copyright (C) April, 2012 by Bob Day
All rights reserved.

When I learned to read in grade school when I was a kid, the method for teaching reading in the school I went to was "phonetics".  You learned the various sounds each letter could make, which was often influenced by its position in the word or by its relationship with other letters (for example, the "bossy 'e'").  I learned to sound out the letters and say the words in my head.  And, throughout my life whenever I read I hear this voice in my head saying the words.  My mother had a thing about phonetics.  To her, that was obviously the only logical way to learn to read.  She used to harangue the teachers at my school about that.  It was obvious to me too.  There was also another method.  At that time it was called the "sight method".  Maybe it still is.  You learned to read by how words looked — internally and by their outline.  I remember a seeing book that taught that method in one of my early grade school classes.  It had pictures of rectilinear contours drawn around the words — in red, I think.  But that was nuts; it didn't make any sense.  Phonetics was the only logical way: letters make sounds; you sound out the words; you can speak the words out loud or say them in your head.  Makes total sense.


Recently, I've found out that the "sight method" also makes total sense.  My daughter learned to read at a very young age.  I think she began to teach herself to read when she was around two years old — long before she went to any kind of school.  Maybe that's significant.  I think she began to learn to read before she knew the alphabet or that letters had sounds.  She must have learned to recognize words purely by what they look like.  I had an interesting conversation with her a few months ago, and during that conversation, I found out something absolutely fascinating.  She does not hear a voice in her head speaking the words as she reads.  I was so totally surprised by that, that I questioned her very closely.  I think she was surprised too.  I don't understand how anyone can not have a voice speaking in their head when they read (I've since tried to turn off the voice, and I absolutely can't).  On the other hand, my daughter doesn't understand how anyone can have a voice in their head when they read.

A while ago, in a computer programming newsgroup on the Internet, there was a discussion about the same thing.  Some people in that discussion said they heard a voice; others said they didn't.  One guy said he sometimes heard a voice, but when he did, he "turned it off".

Well, go a whole lifetime and then learn something totally new about something very basic.  Not obvious at all.  Maybe people who hear a voice in their head as they read learned to read phonetically, and maybe people who don't learned via the sight method.  Maybe.  And there's this theory:  Maybe there are two different kinds of ability to read built into the circuitry of our brains, and maybe some people can learn only by one method, some only by the other, and some by either method but usually not both.  Maybe one reason some people have trouble learning to read is that they're taught by the method that is wrong for them.

Bottom line?  If I had it to do over again and I could choose, I think I would pick the sight method.  My daughter can read very fast — she can read a 200 page book in about an hour, and has no problem with comprehension.  I can read a whole lot faster than I speak, but not nearly that fast.  Also, people who don't have a voice in their head when they read, when they're watching a newscast on TV, can listen to the newscaster and read the crawl script at the bottom of the screen at the same time.  My daughter attests to this.  I can't do that.  For me, reading the crawl script blocks out the newscaster and vice-versa.


Weight Loss

Weight Loss
Copyright (C) April, 2012 by Bob Day
All rights reserved.

It began in the late 1980's when I stepped on the scale one morning.  I weighed 175 pounds! — which is a little overweight for my 5' 8" height.  And I said, right then and right there, "That's it!!".  I was in my mid 50's, and for years my weight had been slowly increasing.  And I didn't feed good.  I felt kind of slow, sluggish, bloated, and heavy.  I could see where things were headed.  So — no more!  And I was able to make it stick.

For years my weight went up and down between the high 160's and the mid 150's, but never again did I weigh 175.  But I still wasn't happy with my weight, and around 1997 I set a goal of losing one quarter of my highest weight.  So my goal became to get down to 131 pounds.  Totally arbitrary?  Yes.  Why one quarter of my weight?  I don't know.  But that was my goal — 131 pounds. It took 5 or 6 years.  When I got down to 140, friends would sometimes tell me that I was too thin.  One person was concerned that I might have anorexia.  I ignored them all.  When I got down to 135, that last 4 pounds was very difficult — it took more than another year.  But I finally did it.  And now I've been under 130 for a couple of years.  I have stayed between 125 and 130, a few pounds above underweight.

Here are some things I learned:
1)  Gaining weight after middle age is not the inevitable process that the therapy world have you believe it is.  You don't have to gain weight if you don't want to.

2)  Weight loss gets easier as you go along.  Your diet just becomes the way you eat  — part of your normal life style.

3)  Calories aren't everything — Calories are the only thing!  If you burn more calories than you take in, you will lose weight.  Exercise has a whole lot of benefits, but it isn't necessary for weight loss.

Diet advice:
1)  Three meals a day; no snacks, and no "in between" meals.

2)  Give up sugar.  No sugar in coffee, soda, or on cereal.  Give up fruit juice — it's mainly just another form of sugar.  Water is the only liquid you need.

3)  No alcohol.  Alcohol has no food value, alcohol is just empty calories.

4)  Measure the amounts of foods you eat by weighing them on a scale.

5)  Establish a very regulated diet that controls the calories you take in.  It's much easier to follow a diet that doesn't require you to be constantly counting calories.

6)  Round out your diet with supplements for nutrients that your diet does not contain enough of.

Complete Protein Organic Whole Grain Bread Recipe

It took me many trials and many adjustments to develop this recipe for a nicely rising whole grain bread.  Note: The amounts may look over obsessive in their precision, but they are just the amounts I somehow ended up with over many trial loaves.  They probably don't need to be all that exact.

Dry Ingredients                                                    Liquid Ingredients
78g  King Arthur organic whole wheat flour         322g   water
        (not stone ground)                                        29g     clover honey (or other mild
79g  organic whole amaranth flour                       1/2 Tbsp  extra virgin organic olive oil,
        (not stone ground)                                                       first cold press
79g  organic whole quinoa flour (not stone ground)
79g  organic whole buckwheat flour (not stone ground)
70g  Bob's Red Mill vital wheat gluten flour
23g  King Arthur Baker's Special Dry Milk (a nonfat non-instant dry milk)
7.7g salt (not sea salt)
6.1g Fleischmann's bread machine yeast

1.  The wheat, amaranth, quinoa, and buckwheat should add to 315g — measure these without rezeroing the scale: 78g, 157g, 236g, and finally 315g.

2.  For baking the bread: In my Zojirushi BBCC-X20 bread machine I used the Home Made baking course in Memory 1, which I set to the following numbers of minutes: Preheat 30, Knead 23, and Rise1 45.  Then I  transferred the loaf to a 8 1/2" x 4 1/2" loaf pan and did a final rise in the bread machine for about 44 minutes (a "Rise2" so the surrounding temperature would be 82.4 degrees F).

3.  Put the loaf into an oven preheated to 425 degrees F and immediately lower the temperature to 400 degrees F. After 9 minutes lower the temperature to 350 degrees F.  Tent the loaf after 19 minutes and continue baking until the internal temperature of the loaf is 201 degrees F (94 degrees C).  The total baking time should be about 34 minutes.

4.  Makes about a 1 1/2 pound loaf.

Free Will

I have been reading Sam Harris's latest book, "Free Will", in which he
claims that free will does not exist.  Basically, he refers to some scientific
studies that show that all actions that you consciously decide to take
have actually been decided for you, totally unconsciously, in your
unconscious mind a few hundred milliseconds, or sometimes a few
seconds, before it enters into your conscious mind and you "decide" to
take it.  But since your unconscious mind has already decided, your
conscious decision is an illusion — you actually had no free will about it.

Harris is aware of the idea of quantum uncertainty, and acknowledges
that Martin Heisenberg, a biologist, has found that certain processes in
the brain occur quantum randomly (thus truly randomly), but says that
has nothing to do with free will.

But I think it may.  Consider this example, which is based on an actual
experience I had in solving a problem:  I had previously noticed that
when you heat a UPS label on a package with a hair dryer and peel it
off, it leaves a sticky area where the label was.  My problem was to
figure out how to get rid of the stickiness.

I don't know much about how the brain works, but as far as I know, there
is nothing to contradict the hypothetical mechanism I will suggest.  And
it coincides with my thinking process in solving the problem:  When you
are trying to solve a problem, perhaps there is an unconscious
mechanism in the brain that generates possibilities at random,
randomly cranking through and randomly associating past events,
objects, tools, materials, devices, et cetera that you know about.  And
when it finds an association that might relate to your problem, presents
it to your conscious mind for further consideration.  In my case of how
to get rid of the stickiness here's how it might work:

My thinking process:
How to get rid of the stickiness???  — think… think… think… (unconscious
mind cranking out a possibility).  Conscious mind:  Aha!  Try rubbing it with
a white pencil eraser.  Hmm — no, that sounds like it would just rub the
sticky around, but not get rid of it.

So,   — think… think… think…
(unconscious mind generating another random possibility).  Conscious
mind:  Aha!  Try spreading glue over the stickiness and letting it dry. 
Hmm — Maybe.  I have some Elmer's white school glue, I'll try that. …
No — sorry — it didn't work.

Back to   — think… think… think…
(unconscious mind generating yet another random possibility). 
Conscious mind:  Aha!  Try rubbing over it with a wax candle.  Hmm —
interesting.  Strange, but who knows?  So try it. … Yes!  Amazing.  It
worked perfectly!  Problem solved.

That's a pretty good description of how my thinking went in my
conscious mind.  Of course, I have no idea what actually went on in my
unconscious mind.  But where do these ideas that just pop into your
head come from?  Especially the one about the wax candle.  The few
candles I even own have been sitting on a back shelf for several years,
untouched.  I haven't even the vaguest (conscious) clue about where
that idea came from.

But, if there is a part of your unconscious mind that randomly
generates ideas for your conscious consideration and decision as to
whether or not to act on, I think that would be an example of free will.

Derivation of the Formula for Mortgage Payments

Derivation of the Formula for Mortgage Payments

Copyright (C) April, 2012 by Bob Day. All rights reserved.

A while ago, I wanted to know the reasoning behind the amount of my payments on my home mortgage. I looked in some accounting books, and all they gave was the formula. None of them gave the rationale behind it. So I sat down and derived it myself. It's not too hard.

Say we borrow an amount "A" at an interest rate of "r" per payment period. (If the payments are made monthly, "r" is the annual interest rate quoted by the bank divided by 12.) We pay back the loan in "N" payments or periods.

For example, for a 30 year mortgage on which payments are made monthly, N would be 30×12 or 360. After N payments, each of the amount "P", the loan is paid off and the amount we owe is reduced to zero.

Derivation of the Formula
The Initial amount we owe is A. At the end of the first payment period, the amount we owe has increased by rA, one payment period of interest, and we make a payment, P. So the total amount we owe after one period is
: A + rA – P, or A(1 + r) – P. We note that this A(1 + r) – P is not only an amount, but also an operator; that is, given the amount of principal outstanding at the beginning of any period, we can apply it to determine the amount of principal remaining at the end of the period.

So applying the operator A(1 + r) – P to the amount A(1 + r) – P remaining at the end of the first period, we get (A(1 + r) – P)(1 + r) – P as the amount remaining at the end of the second period. Similarly, at the end of the third period, the amount of principal remaining is: ((A(1 + r) – P)(1 + r) – P)(1 + r) – P. After N periods (applying the operator and then expanding), the amount remaining will be:

A(1 + r)^N – P( (1 + r)^(N-1) + (1 + r)^(N-2) + (1 + r)^(N-3) + … + 1 ) = 0  [Equation 1].

It equals zero, because after N periods the loan is paid off. Considering just the (1 + r)^(N-1) + (1 + r)^(N-2) + (1 + r)^(N-3) + … + 1 portion, we can reverse the order of its terms and rewrite it as: 1 + (1 + r) + (1 + r)^2 + … + (1 + r)^(N-1)

Representing this series by "S", and letting "R" equal (1 + r), we get:
S = 1 + R + R^2 + … + R^(N-1)
So, RS = R + R^2 + … + R^(N-1) + R^N
Subtracting: S – RS = 1 – R^N, and so S = (1 – R^N) / (1 – R)
Inserting this value for S back into Equation 1:

A(1 + r)^N – P( (1 – R^N) / (1 – R) ) = 0

Finally, Solving for P, the amount we pay each period, we get:

P = rA / (1 – (1 + r)^(-N))

I checked this formula against my own mortgage amount and payments and it agreed exactly! Voila! For example, for a 30 year mortgage for $300,000 at an interest rate of 6% per year paid monthly, the parameters are: A = 300000 (the mortgage amount) r = 0.06 / 12 = 0.005 (the monthly interest rate) N = 30 x 12 = 360 (the number of payments) And the monthly payments would be: 0.005 x 300000/(1 – 1.005^(-360)) = 1798.65 dollars per month.

Another Way: Approximation with a Differential Equation
We can also use a differential equation to get a very close approximation of the payments on a mortgage.  A while ago I was trying to figure out how long it would take a bug walking along a stretching rubber band to get to the end.  After I solved that problem, it occurred to me that the problem of mortgage payments could be solved in a similar way.  It's a nice example of how a differential way of thinking can be used to solve a real-world problem.  Perhaps many problems in finance and economics can be solved using a differential approach.

We start by looking at how fast the mortgage is being paid off:  We now let A(t) be the amount we owe on the mortgage as a function of time.  Note that A(0) is the amount of the mortgage, the amount we borrowed.  Each month, the bank adds an interest amount of r * A to the mortgage and we make a payment of P.  Consequently, the amount we have remaining to pay on the mortgage changes by: dA/dt = r * A – P each month. 

To solve this equation for A requires a little bit of mathematical gymnastics, but it's strictly cookbook.  It can be very easily solved by entering "solve (dA/dt = r * A – P)", without the quotes, into WolframAlpha at www.wolframalpha.com and clicking on the = sign. 

The solution is: A(t) = P/r + C e^(rt), where C is a constant we need to evaluate. After a time T, the mortgage will be paid off, so we have: A(T) = P/r + C e^(rT) = 0. Solving for C, we get, C = -P/r e^(-rT). Replacing C in the solution, A(t) = (P/r) (1 – e^(r (t-T)). So, A(0), the amount of the mortgage (the amount we borrowed) is: A(0) = P/r (1 – e^(-rT)). And finally, solving for P we get: P = r A(0) / (1 – e^(-rT)). For A(0) = 300000 dollars, i = 0.06 / 12 = 0.005 percent per month, and T = 360 months, we get: P = 1797.05 dollars per month, very close to the amount we calculated before.  (But, of course, not quite good enough for the bank!)