Category Archives: measurement

The Monty Hall Problem

[check out my more recent Monty Redux for, perhaps, a clearer exposition]

The Monty Hall Problem illustrates an unusual phenomenon of changing probabilities based upon someone else’s knowledge. On the game-show Let’s Make a Deal the host, Monty Hall, asks the contestant to choose one of three possibilities – Door One, Two or Three – with one door leading to a prize and the other two leading to goats. After the contestant selects a door, another door is opened, one with a goat behind it. At this point the contestant is allowed to switch the previously selected door with the remaining (unopened) door.

Common intuition is that this choice does not present any advantage because the probability of selecting the correct door is set at 1/3 at the beginning. Each door has this 1 out of 3 chance of having a prize behind it, so changing which door you select has no effect on the outcome.

In hindsight, this intuition is wrong. If you initially selected the first goat and then switch when you get a chance, you win. If you selected the second goat and switch, you win. If you selected the prize and switch, you lose. Therefore if you switch, you win 2 out of 3, whereas if you do not switch you win only 1/3 of the time.

So what has gone horribly wrong here:

  1. Why is most everyone’s intuition faulty in this situation?
  2. How does switching doors make any difference?
  3. When did the 1/3 probability turn into a 2/3 probability?

At the beginning of the game you have a 2 out of 3 chance of losing. Likewise the game show has a 2 out of 3 chance of winning (not giving you a prize) at the beginning of the game. Both of these probabilities do not depend upon which door the prize is behind, but only upon the set-up of a prize behind only one of three doors. For instance, an outside service (not the game show) could have set everything up such that both you and the game show would be kept in the dark: there would still be 2 goats and a prize, but neither you nor the game show would know which door led to the prize.

Now imagine that it is the game show that is playing the game. The game show is trying to win by selecting a goat. From this perspective, whichever door that was chosen is good: this door has a 2 out of 3 probability of being a winner (being a goat). Therefore when given the opportunity to change (after the outside service opens a door and shows a goat), there is no reason to do so.

Of course you, the contestant, are the one making the selection, and you do not want a goat. However, if you imagined yourself in the position of the game show at the beginning, as trying to select a goat, you would reasonably assume that, just as the game show did, you were successful in choosing a goat. When given the choice to switch, now that the other goat has been removed, it seemingly makes sense to change your selection.

In this case the easiest way to view the situation is in terms of how to lose, or by considering all the possible outcomes (as mentioned above). Though this is a guess, it seems that our first blush reaction to this problem is always to view it in terms of winning and this is the reason we do not immediately recognize the benefit in switching. We start out with a 1/3 chance of winning and switching doors doesn’t immediately seem to increase this percentage.

To answer how switching doors makes a difference we need to look more closely at the doors. The door that was initially selected has a 1 out of 3 chance of being a prize, and this does not change. If you were to play many times and ignore changing doors, then you would win 33.3% of the time. At the outset the other two doors each have the exact same chance of being a winner, 1 out of 3. So the other two doors combined have a 2 out of 3 chance of containing a winning door.

Now the game show changes the number of doors available from 3 to 2, with one door guaranteed to contain a prize. If you were presented this situation without knowledge of the previous process, then you would rightly put the chance of selecting the prize at 1 out of 2, 50%.

However, you know something about the setup: The door that was initially selected had a probability of having a prize behind it set at 1 out of 3. The thing behind the other door, though, has been selected from a stacked deck: Whatever is behind the door was selected from a group of objects with a 2 out of 3 chance of containing a prize (1/3 + 1/3). You know that the odds on this door are stacked in your favor because the game show knowingly reveals the goat: In the 2/3 case in which you have previously selected a goat, the prize is behind one of the other two doors. When the game-show reveals (and removes) a goat, it guarantees that the prize is behind the last door. Therefore switching doors at the end is equivalent to combining and selecting the probability associated with the two doors not initially selected.

If the game show did not knowingly reveal the goat, you would not be able to take advantage of the stacked deck. Imagine that you select the first door and then another door is opened randomly, revealing a goat. By randomly eliminating this door (and not looking behind the unselected doors) the door that was initially selected becomes unrelated to the present choice: Only by looking behind the unselected doors does the initial selection become fixed in reference to the other doors. Since no one looked behind the doors, some bored, but not malicious, demon could have come and switched whatever was behind the selected and remaining door and neither you nor the game-show would be able to tell. Therefore switching doors when a goat is randomly revealed provides no advantage because the initial selection cannot be related to the probable location of the prize.

Only when the contestant can fix the probable locations of the prize because the location of the prize is known by the game-show, is it possible to assign interdependent probabilities on the location of the prize and the previous selection made. The odds are then tilted in the contestant’s favor by switching away from the low probability initial selection to the door that has the combination of remaining probabilities.


The logic of this needs to be represented game-theoretically with the different quantifiers representing different players of a game of incomplete information. The game would run* like this:

Domain={prize, goat, goat}

Contestant Game Show
1. - ∃x∃y∃z∀a/x,y,z∃b∀c/x,y,z(a=x & b=y & c=z)
2. - ∃y∃z∀a/x,y,z∃b∀c/x,y,z(a=g & b=y & c=z)
3. - ∃z∀a/x,y,z∃b∀c/x,y,z(a=g & b=g & c=z)
4. ∀a/x,y,z∃b∀c/x,y,z(a=g & b=g & c=p) -
5. - ∃b∀c/x,y,z(p=g & b=g & c=p)
6. ∀c/x,y,z(p=g & g=g & c=p) -
7. ∀d∀c/x,y,z(d=g & g=g & c=p) -
8. ∀c/x,y,z(g=g & g=g & c=p) -
9. (g=g & g=g & p=p) -

Line 1 is the initial setup of the prize game: the goal is for the contestant to make his or her placement of the prize and goats match the game show’s placement. Whatever is on the left side of an = will be what the contestant thinks is behind a door and what is on the right of an = will be what the game show puts behind the door, such that each = represents a door. If the formula is satisfied then the contestant will have successfully guessed the location of the prize.

Lines 2, 3 and 4 represent the results of the Game Show placing the prize and goats. Line 5 is the result of the first move of the contestant choosing where he or she thinks the prize is: the ‘a/x,y,z’ means that whatever placed in spot a has to be done independently, i.e. without knowledge, of what x or y or z is. Then the game show reveals a goat behind one of the doors not selected by the contestant. Line 7 represents the choice that is given to the contestant to switch his or her initial placement of where the prize is. Line 8 is the important step: since the contestant does not know what is behind the doors (c/x,y,z) it looks as if there is no advantage to switching. However, the contestant does know that when making a choice to reveal a goat in line 6 that at this point the game show had to know what was behind every door. This means that c is dependent upon b which was depended upon x, y, and z. With this knowledge the contestant can figure out that there is an advantage to switching because the selection of b in line 6 fixed the locations of the prize & goats and in doing so fixed the odds. Since the odds were intially stacked against the contestant, switching to the only remaining door flips the odds in the contestant’s favor, and is done so in this example. Line 9 shows that all the contestant’s choices match up with what the game show has placed behind the doors and hence she or he has won the prize.

* To do a better representation would require keeping the gameshow from not placing a prize anywhere by using a line like ‘x≠y or x≠z’. For graphical brevity I left it out.

Posted in game theory, independence friendly logic, logic, measurement, philosophy. Tagged with , , , .

Positive and Negative Biological Time

In my biorelativity series I used mutations per generation as a measurement of distance. However, with my recent historical/generative musings, specifically the post on the logical foundations of biorelativity (the logic of which is at the foundation of how I arrived at biorelativity), I fear I may have ignored the distinction between a mutation and an adaptation.

Consider an organism with some feature. The feature can be considered both a mutation or an adaptation depending on what the organism is being compared to. If the organism is being compared to another organism, then the feature is likely to be called a mutation. If the organism is being discussed in reference to the ecosystem, then the feature will be referred to as an adaptation.

Now I am sure that there may be some technical properties/definitions having to do with genetics or whatnot that distinguish mutation and adaptation. This is not my concern, though, because in my arguments the two can be used interchangeably.

What does concern me is that there are different sets of related concepts associated with the two words. An adaptation is, to my ear, always a positive thing. A mutation can be good or bad, e.g. mutant freak. By this line of thought adaptations are useful mutations, a subset.

Since mutation is the measurement of time and adaptation is only those mutations which are useful, then we can use adaptation to signify the forward motion of biological time (and forward change of a species as adaptations per generation) which will almost always be what people are discussing (“as time marches on, as things adapt…”). Conversely, to describe biological time going backwards, we could say something like ‘unmutating’.

——

On a slightly different note it is interesting that that there is no word for adapting in the opposite direction: it’s a significant gap. Unadapting? This could imply mere stagnation; the idea here is to think of what it would mean to be adapting in a way to specifically undo previous adaptations. I think a word like this does not nor cannot meaningfully exist: the logical/grammatical structure of adaptation presupposes forward progress.

Consider, “If there were a verb meaning ‘to believe falsely’, it would not have any significant first person present indicative.” (Philosophical Investigations Part II Section x)

“The species is currently *counteradapting*” — It just makes no sense.

Posted in biology, evolution, logic, measurement, philosophy, science, time, wittgenstein. Tagged with , , , , , , , , , .

The Logic of Biological Relativity [draft]

How can we represent biological relativity in logical notation?

Organism a is adapting relative to organism b

Aab

Organism b is adapting relative to a

Aba

Organisms a and b are adapting relative to each other

Aab & Aba

This schema is unsatisfactory because it describes the situation from an indeterminate outside perspective: a and b are said to be adapting relative to each other without regard to the observer describing the situation. Relativity applies to all the perspectives in question (with special focus on any observer perspective) and hence we need a way to include the observer perspective. This means we need to take into account how the observer is adapted such that the observer(s) can be compared to the organisms in question.

To remedy this problem let quantifiers range over organisms and include witnesses to identify the specific organisms in question:

For any organism x, for any organism y, there exists an organism z and there exists an organism u such that x is adapted relative to y according to organism z, and y is adapted relative to x according to organism u.

(∀x)(∀y)(∃z)(∃u)A[xyzu]

Unfortunately this formulation is insufficient because witness z is logically dependent upon both x and y (as is u as well) and we want z to only witness x and u to only witness y: as both z and u are dependent upon both x and y, both x and y must be chosen before selecting z and u. This means that organisms x and y are selected (logically) independent of the witness organisms defeating the purpose of having those witnesses.

Getting around this difficulty is not trivial in first order logic. There is no way in first order logic to linearly order the four quantifiers such that z only depends on x and u only depends on y (Kolak & Symons p.249 [p.40 of the pdf]). Independence Friendly logic suffices though :

(∀x)(∀y)(∃z/∀y)(∃u/∀x)A[xyzu]

This statement says that for any organism x, for any organism y, there exists an organism z that does not depend on y and an organism u that does not depend on x, such that organism x as witnessed by z, and organism y as witnessed by u, are adapted relative to each other.

However, though this statement gets very close to describing biological relativity, if we consider how the witnesses witness the organisms, i.e. how z witnesses the organism x, there is a problem. By stating that z witnesses x and that z is independent of y, the statement ‘x is adapted relative to y as witnessed by z’ is nonsense: since z is independent of y it could not be a witness to ‘x adapting relative to y.’ Likewise for u.

The solution is simple enough though:

(∀x)(∀y)(∃z/∀x)(∃u/∀y)((x=z) & (y=u) & A[x,y])

By letting x=z, making z independent of x and dependent on y, z witnesses y from the perspective of x without requiring x to be chosen before z. Likewise for u: if y=u, u is logically independent of y and u is dependent on x, then u may be chosen before y, u is dependent as a witness to the choice of x and witnesses x from the perspective of y. Perhaps more prosaically: x and y are adapting relative to each other, as witnessed by organisms z and u (who have the equivalent adaptations respectively to x and y), and it is not necessary to predetermine what those adaptations are.

Posted in biology, evolution, fitness, game theory, independence friendly logic, logic, measurement, Relativity, science. Tagged with , , , , , , .

Relativity in Biology notes from 2005

It’s always interesting to see the start of ideas. Although I don’t have anything from the Spring of ’04 when I recall realizing biorelativity for the first time, I have found a file with a ‘last modified’ date of June12, ’05, the contents of which are below:

Quantum Biology

biology: the study of the physical attributes of life.

the rate of mutation is constant, much as the speed of light

organisms mutate. light shines. hence organisms bend/curve life-time as objects bend/curve space-time. greater the mass, the more the curve… the greater the inertia (momentum), the greater the curve. so what is meant by inertia in biology (or in physics)? what does mutation light, as photons light objects? [mutation is the smallest unit of life. photons smallest things with momentum.] we use mutation to view changes of a species. so if a species remains the same, its genetic(?) inertia/ momentum is remaining constant. that with the greatest inertia/ momentum creates the most gravity. that with the greatest inertia/ momentum creates biological gravitation towards itself…

space as vacuum for objects, DNA as vacuum for mutations. objects bend space; mutations do what to DNA? organisms bend life. as objects move to the speed of light their mass (apparently) goes to infinity. as organisms move to the rate of mutation (sex), their DNA (apparently) goes to infinity. as objects slow to absolute 0, their mass (apparently) disappears; as organisms cease mutation (death) the DNA (apparently) disappears. [space is a non-material object, same as concepts, numbers, words etc]

so when there is some massive change to the organism.. say when bats developed sonar, every other mutation became pulled closer around that as to become a part of it. nose, ears, face… eyes are just satellites now

we can then use the fossil history to see what was a major mutative innovation of the day- when preexisting mutations became reoriented around a new mutation (as we can see objects by the change they cause in the motion of other objects, and know their relative size)

location * momentum </= const
species * mutation </= const

——————-

Biological General, Special and plain Relativity in both physics and biology are all confused and mixed together and I was nowhere near my current understanding of biological mass (which didn’t happen till sometime in September of this year and perhaps I’ll go through how I came to that a bit later). It looks like I used DNA for biological mass.

Still, there is a lot of good stuff here.

Posted in biology, evolution, fitness, General Relativity, measurement, philosophy, physics, Relativity, science, Special Relativity. Tagged with , , , , , .

The Logic of Relativity [draft]

How can we represent relativity in logical notation?

a is moving relative to b

Mab

b is moving relative to a

Mba

a and b are moving relative to each other

Mab & Mba

This schema is unsatisfactory because it describes the situation from an indeterminate outside perspective: a and b are moving relative to each other without regard to the observer describing the situation. Relativity applies to all the perspectives in question (with special focus on any observer perspective) and hence we need a way to include the observer.

To remedy this problem let quantifiers range over perspectives and include witness individuals to identify the specific perspectives in question:

For any perspective x, for any perspective y, there exists a perspective z and there exists a perspective u such that x is moving relative to y according to witness z, and y is moving relative to x according to witness u.

(∀x)(∀y)(∃z)(∃u)M[xyzu]

Unfortunately this formulation is insufficient because witness z is logically dependent upon both x and y (as is u as well) and we want z to only witness x and u to only witness y: as both z and u are dependent upon both x and y, both x and y must be chosen before selecting z and u. This means that perspectives x and y are selected independent of the witness perspectives defeating the purpose of having those witnesses.

Getting around this difficulty is not trivial in first order logic. There is no way in first order logic to linearly order the four quantifiers such that z only depends on x and u only depends on y (Kolak & Symons p.249 [p.40 of the pdf]). Independence Friendly logic suffices though :

(∀x)(∀y)(∃z/∀y)(∃u/∀x)M[xyzu]

This statement says that for any perspective x, for any perspective y, there exists a perspective z that does not depend on y and a perspective u that does not depend on x, such that perspective x as witnessed by z, and perspective y as witnessed by u, are moving relative to each other.

However, though this statement gets very close to describing relativity, if we consider how the witnesses witness the perspectives, how z witnesses the perspective x, there is a problem. By stating that z witnesses x and that z is independent of y, the statement ‘x is moving relative to y as witnessed by z’ is nonsense: since z is independent of y it could not be a witness to ‘x moving relative to y.’ Likewise for u.

The solution is simple enough though:

(∀x)(∀y)(∃z/∀x)(∃u/∀y)((x=z) & (y=u) & M[x,y])

By letting x=z, making z independent of x and dependent on y, z witnesses y from the perspective of x without requiring x to be chosen before z. Likewise for u: if y=u, u is logically independent of y and u is dependent on x, then u may be chosen before y, u is dependent as a witness to the choice of x and witnesses x from the perspective of y. Perhaps more prosaically: x and y move relatively to each other, as witnessed by z and u (who have the equivalent perspectives, respectively to x and y), and it is not necessary to predetermine what those perspectives were.

A time variable rounds everything out nicely:

(∀t)(∀x)(∀y)(∃z/∀x)(∃u/∀y)((x=z) & (y=u) & M[t,x,y])

So, at time t (say now) let’s let u be your (the reader’s) perspective and z be my (the author’s) perspective. Then this statement describes our current motions as relative to each other because my perspective depends upon y, which is your perspective and your perspective depends on x, which is my perspective. Success!

——————————

Posted in game theory, independence friendly logic, logic, measurement, physics, Relativity, science. Tagged with , , , .

Why Evolutionary Principles Cannot be Used to Support Racial Prejudices DRAFT

Evolutionary principles are sometimes used to justify racial prejudices. While no rigorous scientific study has yet proven one race to be inferior to any others it should be recognized that it is in principle impossible to prove racial superiority/inferiority and hence no study ever will.

Firstly a note on the meaning of ‘more evolved’ and ‘less evolved’. Every species on the face of the earth today has been evolving for the exact same amount of time. We all started at the same time. However you believe that life started, either as a single celled organism in the ancient seas of earth or by some intervention, if you believe in evolution then everything started roughly at that one point and proceeded from there. Hence we are all equally evolved, from humans to gnats. The only things that may be considered less evolved are things that are no longer around to complain about it.

Secondly, if the claim of racial superiority/inferiority is not one of being ‘more evolved’, then it is a claim about being differently evolved to have some properties that other races do not have. This roughly means that one race has some qualities that make them more fit, or, conversely, one race lacks some features that the rest of us have (even if they have made it this far) that makes them less fit. Either way the claim boils down to either having or lacking a certain characteristic or characteristics. These characteristics, by definition, were passed down through the successive generations eventually proliferating throughout a family, later a population and thence to the entire race some time later.

In order to objectively measure the fitness of an organism or species we need to be able to replicate a controlled environment and a control group. Regardless of the implications of cloning people for a control group, the concept of a controlled human environment will present us with insurmountable theoretical problems.

The environment that we place organisms in is the ‘test’ that we are judging them on. Specifically, if we want to see which of two species flourishes in a particular environment we would place both in the same environment and see what happens. However, not only is it impossible for us to replicate an environment to test people in, none of us know what the future will hold for our species. Hence, without foresight into the future, it is impossible for us to have an environment that could be used as an objective test environment.

In lieu of this impossible situation, approximations are the only possibility. To approximate requires making decisions about what will be included and what will be excluded. The decisions made will influence the results making them a function of the decisions made. Hence it is impossible to approximate without biasing the results, rendering the study useless for the purpose of establishing superiority.

Since we are all here now and none of us knows the future, there will be no study that can prove the superiority or inferiority of any race. Anyone who claims otherwise is claiming they can predict the future perfectly, is racist or both.

——-

If there are any argument structure fans (such as myself woohoo!) this argument’s in a mathematical induction style. The first paragraph argues against a base case (of a race being no more fit than any other) and then the subsequent paragraphs argue against any possible way to argue that the property (of being no more fit) could be used as an inductive hypothesis: Base case is day zero for our species, in which we are all obviously equally fit, and then the induction is on day n (today) and n+1 (tomorrow). Since we are all here now and none of us know what tomorrow holds (by way of Relativity in Evolutionary Biology we have no way objectively view the trajectory of our species), we can move from n to n+1 and the inductive step is made. Hence it is impossible to prove future fitness in our species.

If anyone cares to give me some feedback, I’d like to know if you think it is worthwhile to include some of this argument structure stuff into the body of the paper. I’ve had experiences where I’ve written arguments but people have completely missed them because they were not as familiar with argument forms.

Also would it be worth it to have some commentary on recent developments such as Watson’s gaff or the recent NYTimes article about genetics and race?

Posted in biology, evolution, fitness, measurement, news, philosophy, science. Tagged with , , , , , , , , , .

Evolutionary Drift, revisited yet again

With my recent paper on Measuring Fitness I realize that my previous responses to evolutionary drift, though not incorrect, may have not stated the solution particularly clearly. When fitness is defined and measured as described in the aforementioned article, evolutionary drift is irrelevant. The method of measuring the fitness of an organism or species makes no reference to any mutations whatsoever. Therefore evolutionary drift is no problem for the theory of fitness described here.

If we are trying to identify whether a certain mutation makes an organism more fit, we can of course test it against an organism without that mutation. However if we are unable to test it (say we are studying a historical period or it is just unfeasible), then I believe my previous posts are accurate. I mainly argue that you can’t tell what exactly makes an organism more fit- it’s an underdetermination thesis of sorts – based upon our limited evolutionary perspective.

I think I just failed to say how irrelevant drift was to fitness before this.

Posted in biology, evolution, fitness, General Relativity, measurement, philosophy, physics, Relativity, science. Tagged with , , , , , , .

Measuring Fitness

The basic premise is to measure fitness in a conceptually similar way to how we measure mass.  To measure mass we can use a scale to compare the effect of gravity on a test object to an object with an agreed upon mass, or we can compare the test object’s resistance to acceleration as compared to an object with an agreed upon mass.  These methods measure the ‘gravitational’ mass and ‘inertial mass’ respectively.

Gravitational Mass and Selection Fitness

Measuring an object’s gravitational mass requires a uniform gravitational field, e.g. the gravitational field at the surface of the earth.  The gravitational field accelerates things based upon how massive they are: the more massive an object is the greater the force that the gravitational field exerts.  To measure the mass of an object it is placed on one pan of a scale and pre-calibrated masses (objects of known mass) are placed on the other pan.  When the two pans are level the test object has an equivalent mass to the calibrated masses because they have equivalent forces being applied to them by the gravitational field.

To measure fitness we require a similar experimental setup.  First, a uniform gravitational field: according to the General Theory of Biological Relativity ecosystems create large natural selection fields.  A uniform natural selection field requires an ecosystem free from disturbances which could skew the reproductive rates of the organisms.  Secondly we would need organisms with a standard fitness.  A suitable organism would be easily clonable and of a fitness that we suspect our test organism to be near.  That organism’s fitness would be defined as one ‘biogram’ (or what you will).  Lastly we would need to see how the organisms fair in the ecosystem.  Their fitness will be proportional: if both proliferate (or die off) at the same rate, then their fitness will be equivalent, if one does much better than the other then it’s fitness will be proportionally higher.

Inertial Mass and Survival Fitness

Measuring an object’s inertial mass is measuring how resistant it is to acceleration as compared to how resistant to acceleration an object of known mass is.  To measure inertial mass the test mass is attached to a spring clamped horizontally to a stable structure.  The mass and spring are then pulled to one side and let oscillate back and forth: the more massive the object, the slower oscillations.  The number of oscillations per unit of time can be compared to the oscillations per time of a known mass and thence the inertial mass can be calculated.

As above a controlled environment and an organism whose fitness is known (even if by definition) is needed.  However the organisms need to be ‘accelerated’ for this measurement.  According to the General Theory of Biological Relativity environmental conditions will dictate how a species changes over time.  Therefore to ‘accelerate’ a species a changing environment is needed.  Simply put: measuring ‘survival fitness’ is measuring how well an organism or species fairs in a changing environment.  For example a plant that can survive in a wide range of temperatures will be fitter than one that requires a narrow temperature range.  If a test plant proliferates and the benchmark organism withers under a temperature swing, the test organism has a greater fitness.

Posted in biology, evolution, fitness, General Relativity, measurement, philosophy, physics, Relativity, science. Tagged with , , , , , .
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