Difference between revisions of "Proving negatives"
|Line 2:||Line 2:|
: Sometimes you can't prove a truth.
: Sometimes you can't prove a truth.
Latest revision as of 17:02, 24 February 2018
This is a partially revised and somewhat updated version of the original HTML version of this essay.
I am baffled by a claim I hear made in arguments with friends of mine. They say, "You can't prove a negative." This seems to me to be an untenable position on several different grounds.
- 1 I. The claim is self-referentially inconsistent
- 2 II. Formal Logic Does Not Treat Negatives as a Special Case
- 3 III. Disproof by Counter-Example
- 4 IV. Let the Equivocation Begin
- 5 V. Concluding Uncharitable Postscript
- 6 References
- 7 Sources
- 8 Links
- 9 Revision history
- 10 Comments and Corrections Welcome
I. The claim is self-referentially inconsistent
If it were possible to prove that we can't prove a negative, then the existence of the proof would defeat the claim, because we would have proven a negative.
I like the argument from self-referential inconsistency very much. I did my master's thesis on Gaston Isaye, a Jesuit who taught in Belgium and who used "retortion" (French: retorsion) to identify the first principles of thought. He is one of many Transcendental Thomists who used a similar form of argument in their arguments with Kantians. When someone makes a claim whose truth is in conflict with the act of making the claim, an opponent may retort that doing so is an act of intellectual dishonesty, even though there is no strict contradiction in terms. So, for example, if someone were to write, "No one can write an intelligible sentence in English," I would retort: "But you just did. The evidence that you are wrong is given in the assertion you have just made."
In the same way, all of the following assertions are self-referentially inconsistent:
- "There is no truth."
- "No one can know anything for certain."
- "Absolute certainty will always elude us." 
- "Claims that cannot be tested, assertions immune to disproof are veridically worthless, whatever value they may have in inspiring us or in exciting our sense of wonder."
- Note that this is a claim is a claim that cannot be tested and is an assertion immune to disproof, and therefore, according to the standard it sets, is veridically worthless.
- "In science there are no authorities."
- Says who?
- "Physicists do not derive Maxwell's equations or quantum mechanics from scratch. They try to understand the principles and the mathematics, they observe its utility, they note how Nature follows these rules, and they take these sciences to heart, making them their own."
- Any scientist who set out to personally verify the Handbook of Physics and Chemistry would be labeled a fool by his colleagues. No one person could "test everything" recorded there in a single lifetime.
- "All that can be hoped for is a set of successive approximations."
- This is not an approximation capable of further refinement.
- "We are all flawed and creatures of our times."
- This is a statement about all humans at all times and in all places. If it is true, it proves that it is false because it shows that we can transcend the limits of our own time and place.
- "Doubt everything."
- If I am to doubt everything, I must doubt doubting. I must doubt the advice to "Doubt everything." The program of universal doubt implies that we can trust our ability to stop doubting when we have found something worthy of belief--but, in that case, we are not doubting "everything," only some things, while preserving unquestioned faith in our own powers of observation and reasoning.
- "The only things that are true are those which can be verified empirically."
- This claim cannot be weighed, measured or observed directly by the senses or by the use of any laboratory instruments. If it is true, it proves that it is false because it is not verified by any empirical method.
- "The only things that are true are those which can be verified by controlled scientific experiments."
- There is no controlled scientific experiment by which this claim can be verified.
- "The only things that are true are those which have been proven to be true."
- This is an unproven assertion. For me, given my familiarity with retortion, just the observation that the claim "No one can prove a negative" is self-referentially inconsistent is enough to throw it into the dustbin of Stupid Human Tricks. Others do not find the argument from self-referential inconsistency as persuasive as I do. A Jesuit friend says "I hate retortion. It's like picking your own pocket. You are no richer afterward than you were before." I respectfully disagree. It seems to me that retortion keeps us from embracing falsehoods. I consider that a net benefit, even if there is much more to do in philosophy than just not taking untenable positions.
II. Formal Logic Does Not Treat Negatives as a Special Case
In order to evaluate the proposition "no negative can be proven," we have to think for a moment about what we mean by "proof."
I have studied symbolic logic as part of my training in philosophy (M.A. Hum., Fordham University, 1977). There are no special rules for the formation or testing of "negatives." (Aristotelian or traditional or syllogistic logic does distinguish between affirmative and negative propositions. See analysis of categorical syllogisms below.) Formal logic depends on a number of axioms:
- Principle of Identity.
- A thing is what it is and is not what it is not.
- a = a
- a ≠ ~a
- It is a violation of the principles of formal logic to change the meaning of terms in the course of an argument. Doing so is called equivocation. More colloquially, it might be called switching horses in mid-stream. You find that the argument you rode in on is not going to carry you to the desired conclusion, so you (magically and unethically) hop onto a different horse.
- Principle of Non-Contradiction.
- It is a fundamental violation of the rules of logic to assert both a proposition and its opposite. Only one can be true and the other must be false.
- Principle of the Excluded Middle.
- A statement is either true or false; there is no middle ground between true and false. If a statement is true, its opposite is false. If a statement is false, its opposite is true. [I am aware that other systems of logic have been proposed that accept values other than true or false. But if these other systems are proposed as alternatives that disprove the value of orthodox logic, then they form a Boolean pair and we are back in the world of the Excluded Middle--one system or the other is correct, but not both.]
- Principle of Equality and Substitution.
- If two expressions are equivalent, one may be substituted for the other.
- If a = b
and b = c
then a = c
- If a = b
- Principle of Deductive Reasoning.
- If a premise is true, all consequences drawn from it by the application of axioms or theorems are also true. If a premise is false, all consequences drawn from it by the application of axioms or theorems are also false. "Abstraction, formalization, axiomatization, deduction--here are the ingredients of proof."
- Definitions of Affirmation and Negation.
- True and false are taken to be polar opposites. The negation of a truth is a falsehood and the negation of a falsehood is a truth. When a "double negative" is found, the two negative signs may be removed:
- p = ~(~p)
- p = p
- Definitions of Operations (Boolean Logic).
- The definitions of affirmation and negation overlap Boolean Logic, which is at the heart of most of our digital computers. Every component necessary for computation is based on three operations: NOT, AND, and OR.
- NOT 1 is 0
- NOT 0 is 1
- 1 AND 1 is 1
- 1 AND 0 is 0
- 0 AND 0 is 0
- 1 OR 1 is 1
- 1 OR 0 is 1
- 0 OR 0 is 0
- As a matter of symbolic convention, 0 represents false and 1 represents true. Using this convention, the standard rules for syllogisms can be mapped into Boolean logic. Although Boolean logic does not explicitly state or defend the principles of identity, non-contradiction, excluded middle, equality and substitution, or deductive reasoning, they are all employed implicitly. 0 is 0, not 1. 1 is 1, not 0. The conjunction of a true and false statement is false. The conjunction of two true statements is true. Determining "truth" or "falsehood" is not dependent on whether the statement in question is an affirmation or a negation.
None of these principles can be proven to be true because they must be assumed in the course of writing any proof, including a proof of their own validity. That means that any attempt to prove them will invoke them, which would make the proof guilty of assuming its own conclusion, which is also known as "begging the question." That is a vicious circle. As Aristotle noted 23 centuries ago, because anyone who denies one of these principles can only do so by using them, the principles may be defended by retortion. So, for example, it is self-referentially inconsistent to deny the principle of identity because the skeptic must use the principle of identity in order to identify the proposition that is being denied.
Once the basic principles of logic are understood and accepted, there is no limit to the number of systems that can be generated from them:
- "There is a popular saying that knowledge always adds, never subtracts. . . . Mathematics builds on itself; it is aggregative. Algebra builds on arithmetic. Geometry builds on arithmetic and on algebra. Calculus builds on all three. Topology is an offshoot of geometry, set theory, and algebra. Differential equations builds on calculus, topology, and algebra. Mathematics is often depicted as a mighty tree with its roots, trunk, branches, and twigs labeled according to certain subdisciplines. It is a tree that grows in time."
In 1981, Davis and Hersh estimated that there are over 3000 varieties of mathematics and that about 200,000 theorems (statements proven to be true) were being published each year.
Formal logic depends upon correct translations from ordinary language.
- This is the hard part of using symbolic logic to check the validity of a proof. So far as I know, there are no translating machines available (an asserted and unproven negative--just reporting the state of my brain this morning). I appeal to an authority in support of my opinion: "The identification of fallacies involves an understanding of language and an ability to 'read between the lines' that is beyond the capacity of current computer software."
- When I studied formal logic at Fordham lo! these many years ago, we were given "word problems" designed for the purposes of the course that resembled the sort of problems given in algebra courses. Both in my courses in algebra and in my course in formal logic, I noticed that not all of my fellow students were equally adept at making these translations.
- There is no special rule for translating negatives. Any proposition can be represented by "P", whether its content is syntactically positive or negative. The contradiction of the proposition is then represented by the NOT operator (~ in typescript): if P represents the proposition in question ("There are no proven negatives") then ~P represents the opposite position ("There are proven negatives"). But for the purposes of testing the validity of an argument, we could let P represent "There are proven negatives" and ~P stand for "There are no proven negatives." It makes no difference, so long as the translation is made consistently at the beginning and the end of the formal analysis.
- The principles of identity, non-contradiction and excluded middle prohibit the assertion of both P and ~P. If P is true, ~P is false; if ~P is false, P is true. This is just the meaning of the definitions and axioms that give formal logic its mojo.
- The denial of a consequence can lead to the denial of a premise. If a proposition P implies Q, and we find that the case is ~Q, we may conclude ~P. The name of this form of argument is modus tollens:
- Major premise: If I had set my pants on fire, I would feel the heat.
- Minor premise: I do not feel any heat.
- Conclusion: I have not set my pants on fire.
- In the real world, especially in an exciting environment, there may be a time lag between the onset of the fire and the recognition of the meaning of the pain. I've read about RC pilots putting a "glow igniter" in their pocket, having it short out against a set of keys or other metallic objects, and taking a little while to realize what was going on. I've learned from their mistakes. I rarely, if ever, put a glow igniter in my pocket. I have not yet set my pants on fire at the flying field. As the stock brokers say, "Past performance is no guarantee of future results," so I am not certain that I never will set my clothing on fire.
Proof depends upon premises.
- "Any mathematical theory such as arithmetic, geometry, algebra, topology, etc., can be presented as an axiomatic scheme wherein consequences are deduced systematically and logically from the axioms. Such a logico-deductive scheme may be compared to a game and the axioms of the scheme to the rules of the game. Anyone who plays games knows that one can invent variations on given games and the consequences will be different. A non-Euclidean geometry is a geometry that is played with axioms that are different from those of Euclid." "All we can say in mathematics is that the theorem follows logically from the axioms. Thus the statements of mathematical theorems have no content at all; they are not about anything. On the other hand, according to the formalist, they are free of any possible doubt or error, because the process of rigorous proof and deduction leaves no gaps or loopholes."
- Mathematics "as a whole" has not been proven to be "true."
- David Hilbert posed the challenge in 1900 to prove that "the axioms of logic are consistent." Bertrand Russell and Charles Whitehead took up the challenge and published Principia Mathematica in 1910-1913. Later Russell said, "I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expect me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable."
- Davis and Hersh reproduce a page from PM with the comment, "Russell and Whitehead were pioneers in a program to reduce mathematics to logic. Here, after 362 pages, the arithmetic proposition 1 + 1 = 2 is established."
- "As it happened, certainty was not to be had, even at this price. In 1930 Gödel's incompleteness theorems showed that the Hilbert system was unattainable--that any consistent formal system strong enough to contain elementary arithmetic would be unable to prove its own consistency. The search for secure foundations has never recovered from this defeat" (D&H, 337). [After reading Torkel Franzén's book, Gödel's Theorem: An Incomplete Guide to Its Use and Abuse, I see that I may have to modify or remove these last three paragraphs. Mathematicians disagree about the MEANING of Gödel's Theorem for mathematics and about the changes in the philosophy of mathematics that should take place because of the theorem. Stay tuned for another revision, but probably not any time soon. My claim that "mathematics as a whole has not been proved to be true" is almost certainly a mathematically meaningless proposition.]
Now let's talk about proving negatives.
- When people introduce the maxim, "You can't prove a negative," they are making an implicit appeal to the following syllogism:
- Major premise: No negative can be proven.
- Minor premise: The proposition being debated is a negative.
- Conclusion: The proposition being debated cannot be proven.
This syllogism is true if and only if the major premise--the assertion--is true. But the assertion is not self-evident. That is, its meaning is not derived from the definition of the terms involved. Nor is it an axiom of formal logic or mathematics. It is not a theorem that has been proven by any logician or mathematician--you can either take my word for this or you can search the literature as I have. I will change my mind and withdraw this page if I can be proven wrong on this point.
III. Disproof by Counter-Example
My claim that "There is no rule about proving negatives in formal logic" is an example of a negative I cannot prove. In order to establish it decisively, I would have to have read and correctly evaluated every text in logic. I have not done so and will not do so. My conviction is based on familiarity with certain types of formal logic which I take to be representative of the whole.
Is my assertion of an unprovable negative self-referentially inconsistent? Have I proven my adversaries case by illustrating their point? No. The negation of the maxim "You cannot prove a negative" is not "All negatives can be proven" but "At least one negative can be proven" or, to mirror the maxim more closely, "You can prove a negative" or, perhaps, closer to the view I defend here, "You can prove some negatives." The existence of one (or more) unprovable negatives does not render my position untenable. One single counter-example destroys the claim that "No negatives can be proven", but no single example or group of finite examples establishes it.
In deciding the issue, we want to weigh the evidence carefully. "'One unerring mark of the love of truth,' wrote John Locke in 1690, 'is not entertaining any proposition with greater assurance than the proofs it is built upon will warrant.'" I strive to meet the standards set by "that remorseless taskmaster called the scientific method: Everything hinges on the matter of evidence." I will alter my opinion if the evidence forces me to do so; if it does not compel me, I will not change my mind.
The maxim in question cannot be established by any finite list of examples of negatives that are difficult or impossible to prove. The maxim covers all negatives of any kind whatsoever. It is an unrestricted proposition. The set of all possible negatives is probably infinite, since there is no reason why new negatives cannot be constructed at will. The resources are readily available to anyone who wishes to create a new negative hypothesis. No matter how may examples of unprovable negatives one lists, if it is a finite list, it is not exhaustive and cannot establish the universal negative. Davis and Hersh describe "another well-verified conjecture--known to be true in the first billion cases--which Littlewood proved is false eventually" (364).
The maxim that "You cannot prove a negative" is defeated by finding one example of a provable negative. I've found that when I start giving examples of provable negatives, people start hopping on different horses from the one they rode in on. Suddenly, their definition of what "a negative" is and what "proof" is start to change and become elusive. It is very frustrating, which is why I've taken so much time to write this essay and map out my position. I'm sure I will collect more examples of equivocation as people read and comment on this. As I give examples, people will say "No, that's not what I mean by proof of a negative." Well, define your terms. I will then give a counter-example according to your definitions. If you can't define your terms, then you're not qualified to advance the maxim as part of reasoned discourse.
A. Examples of Provable Negatives
A friend writes: "We live in a world that is easier to prove something could not happen, than the reverse." That seems to me to be a brilliant insight. Lots of my examples below fit that pattern.
"There is no elephant in my pocket."
- This is an existential proposition. Some people might argue that from the mere meaning of the words, we could know that it is true, but just to make sure I've looked in my pocket to verify my a priori instincts. There is no elephant of any kind there.
- I'm asking you to take my word for this. I think I can tell the difference between an elephant and pocket lint. You may, if you wish, verify the absence of the elephant for yourself if you will take the trouble to visit me and inspect my pockets firsthand. I trust that, on a good day, you, too, can tell the difference between an elephant and pocket lint.
"No camera can record its own construction."
- Knowing that this is true depends on understanding the meaning of the words "camera" and "construction." A camera is a complex system of parts. The parts in isolation from each other do not form a working camera. The camera cannot record anything until the parts are assembled, and so that very camera cannot take pictures of its own parts being assembled into a whole.
"The muffler is not on my car."
- This was true the day I took our silver Oldsmobile to a muffler shop shortly after the muffler fell off. Both the mechanic and I could see for ourselves that the muffler was not on the car. Fortunately, the man was able to remedy the situation at a very reasonable price and this statement has ceased to be true. But for a little while, it was a very provable negative. I had other evidence besides visual to help me reach the proper conclusion at the time that the observations were made. So I may now write, "The muffler was not on my car between 12:00 noon and 1:00 PM on July 29, 2003." (I've given the bill to the man in charge of cars and the credit card record to the Treasurer, so I don't have the time of day right at my fingertips.)
"Aether does not exist."
- Physicists in the 19th century assumed that every wave needed a medium; sound travels through the air the way waves travel through water. Since they knew that light acted as if it were a wave, they hypothesized that the wave would be carried by the "aether." Maxwell's equations showed that light propagates by the production electro-magnetic fields, which require no physical substratum. Careful observations of the speed of light showed that no effect of the motion of the earth through the aether could be detected.
"It is wrong to think that only ferrous metals are attracted by magnets."
- I had originally proposed that "This is a piece of non-ferrous metal" would be an example of a provable negative, but I was wrong (the three hardest words in the English language for a man to say). I alleged that if a piece of metal contained iron, a magnet of sufficient strength would attract the metal; therefore, if a magnet did not attract the metal, one could conclude that the metal is non-ferrous, without having to determine exactly what kind of metal it is. Bill Hardin, a friend who is also an accomplished electrical engineer, sent me this information:
- In a book (pamphlet really, only 38 pages) originally published in 1951, "Design, Construction & Operating Principles of ELECTROMAGNETS for Attracting Copper, Aluminum and other Non-Ferrous Metals" by Leonard R Crow, the author opens his book with: "Back in 1935 I gave a public lecture demonstration at the physics department of Illinois University in which I used an electrical training aid of my own design and construction to show that an electromagnet could be used to attract non ferromagnetic metals of good electrical conductivity. In the following five years I devoted much thought, time and effort to making various types, styles and sizes... Following is a description of, and basic fundamental operating principles pertaining to the type of electromagnet that I have found most effective for the attraction of non ferrous-metals." Basically, he used a toroidal coil driven to saturation with the addition of a traditional core to achieve the non-ferrous magnet. I've been thinking of trying it, since I have a couple of toroids in my scrap box.
- A reprint of the pamphlet is available. The ad for the pamphlet hints that running AC current through the coil may be part of the magic.
- "Ferromagnetic materials are strongly attracted by a magnetic force. The elements iron (Fe), nickel (Ni), cobalt (Co) and gadolinium (Gd) are such materials." Another site says that "iron, cobalt, [and] nickel ...are intrinsically magnetic metals."
- Metal detectors used induced magnetic fields in non-ferro-magnetic materials to determine what the metal is:
- "Ferromagnetic metals are ones that have intrinsic magnetic structure and respond very strongly to outside magnetic fields. The non-ferromagnetic metals have no intrinsic magnetic structure but can be made magnetic when electric currents are driven through them. Good metal detectors produce electromagnetic fields that cause currents to flow through nearby metal objects and then detect the magnetism that results. Unfortunately, identifying what type of non-ferromagnetic metal is responding to a metal detector is hard. Mark Rowan, Chief Engineer at White's Electronics of Sweet Home, Oregon, a manufacturer of consumer metal detecting equipment, notes that their detectors are able to classify non-ferromagnetic metal objects based on the ratio of an object's inductance to its resistivity. They can reliably distinguish between all denominations of U.S. coins--for example, nickels are relatively more resistive than copper and clad coins, and quarters are more inductive than smaller dimes. The primary mechanism they use in these measurements is to look at the phase shift between transmitted and received signals (signals typically at, or slightly above, audio frequencies). However, they are unable to identify objects like gold nuggets where the size, shape, and alloy composition are unknown."
- Induced magnetic fields in aluminum tracks are part of a design for maglev trains:
- "There are many techniques for supporting a train on magnetic forces, but the simplest and most promising involves electrodynamic levitation. In this technique, the train has a strong magnet under it and it rides on an aluminum track. The train leaves the station on rubber wheels and then begins to fly on a cushion of magnetic forces when its speed is high enough. Its moving magnet induces electric currents in the aluminum track and these currents are themselves magnetic. The train and track repel one another so strongly with magnetic forces that the train hovers tens of centimeters above the track."
- There is an excellent provable negative in the following paragraph (in bold):
- "In most atoms, electrons occur in pairs. Each electron in a pair spins in the opposite direction. So when electrons are paired together, their opposite spins cause there magnetic fields to cancel each other. Therefore, no net magnetic field exists. Alternately, materials with some unpaired electrons will have a net magnetic field and will react more to an external field. Most materials can be classified as ferromagnetic, diamagnetic or paramagnetic.".
- Bottom line: it looks like my original negative needs to be qualified. If a material is not attracted to a permanent magnet of sufficient strength, we may conclude that it is probably not a ferromagnetic material.
"This button is not made of metal and is not as shiny as metal. It would not produce the same kind of gleam in a photograph as this man's two rings do in this photograph. It would be a waste of time to prove this point by photographing the button under controlled conditions in a laboratory."
- This is an example drawn from a recent discussion of a button found on a deserted island in the Pacific by some of my acquaintances. I've seen the button and I agree that we don't have to do the controlled experiment, pace Sagan and other positivists, in order to know how it would appear in a photograph duplicating the lighting conditions of the picture taken of the man with two rings. Note well: Deciding how to allocate scarce resources of time and money is a standard part of scientific research. Not everything needs to be proven by quantification and experiment (another unproven, but I think extremely reasonable negative).
"The sun does not revolve around the earth."
- In the mythic history of science, it is a given that "the Church was wrong to condemn Galileo for teaching that the earth revolves around the sun." (The true history of the Galileo affair is much more complex than that, but I will pass over all of those qualifications in silence.) The observation of stellar parallax in 1838, coupled with Newton's law of gravity, have decisively settled the issue that the sun is vastly larger than the earth and that its gravitational force is the dominant cause of the earth's motion in orbit around the sun, not vice-versa. (Galileo (and Copernicus before him) had no such evidence for their opinion--but they did bet on the right horse in spite of the absence of "perfect proof", and were right to do so, Locke's dictum notwithstanding.)
"No separated magnetic poles (magnetic monopoles) have been observed."
- Electrical charges can be separated--electrons can move around without hauling a proton along with them, or vice-versa. But (so far), it seems that wherever there is a north magnet pole, there is a south pole as well. This is a good example of a 'negative' that makes just as much sense in its 'positive' version: Magnetic poles have always been found in pairs.
- If high-energy physics can bring enough force to bear, it is possible that some day a magnetic monopole will be observed. Folks are looking closely at cosmic rays on the supposition that magnetic monopoles may be part of the radiation mix.
In browsing to understand the next three proven negatives, I found a wonderful article on the disease of trisectionism and an analysis of cranks.
"You cannot trisect an angle using a compass and an unmarked straightedge."
- Anyone who has played with a compass and straightedge knows how hard a challenge this is. Pierre Laurent Wantzel proved the impossibility in 1837 using mathematics that are far beyond my ken.
- Here's the short story from Math Forum:
- "The impossibility proofs depend on the fact that the only quantities you can obtain by doing straightedge-and-compass constructions are those you can get from the given quantities by using addition, subtraction, multiplication, division, and by taking square roots. These numbers are called Euclidean numbers, and you can think of them as the numbers that can be obtained by repeatedly solving the quadratic equation. These three problems require either taking a cube root or constructing pi. A cube root is not a Euclidean number, and Lindemann showed that pi is a transcendental number, which means that it is not the root of an algebraic equation with integer coefficients, making it too non-Euclidean."
"Given an arbitrary circle, it is impossible to construct a square of the same area using only straight edge and compass."
- "A circle and square have an equal area only if the ratio between a side of the square and a radius of the circle equals the square root of pi. Lindemann proved that two line segments cannot be constructed to have lengths in this ratio and therefore this method cannot square the circle."
Given the side of a cube, you cannot construct the side of a cube that has twice the volume.
- "Doubling a cube whose edge equals 1 yields the equation x^3 = 2, whose solution (the length of a side of the larger cube) is the cube root of 2. The problem cannot be solved because the so-called Delian Constant (the cube root of 2) is not a Euclidean number."
"There are no dragons."
- Animals that can breathe fire and fly do not exist and have not ever existed, except in the imagination. The laws of combustion and biology are against the idea that dragons could have or can exist.
"There is no Loch Ness Monster."
- "Using 600 separate sonar beams and satellite navigation technology to ensure that none of the loch was missed, the team surveyed the waters said to hide Scotland's legendary tourist attraction but found no trace of the monster."
"There is no gold mountain."
- There is no scientific reason why a mountain made out of pure gold can't exist. It is simply a matter of fact that there is no such mountain. Fort Knox and similar vaults don't count. They are foothill-sized at best.
"The present king of the United States is bald."
- (I've deliberately gotten away from 'the present king of France' because I fear my American readers may not know enough about France to understand the point.) There are two interesting points about this kind of claim.
- 1. Both the claim and its denial ("The present king of the United States is not bald.") are nonsense because the implicit assertion that there is a king of the United States is false. That there is no king may be proven from a careful study of the documents upon which our nation is founded.
- 2. "Baldness" is a predicate which represents a continuum of hair loss rather than a simple binary condition (bald or not bald). I'm going bald even as I write this, but I've still got lots of hair toward the back of my head and on the sides. This is a job for fuzzy logic, a magnificent and charming development in the last few decades that has plenty of growth potential, even if my hair doesn't. See Carl Sagan's 14th fallacy below as well.
Four-Color Theorem: No map of adjacent regions (i.e. those sharing a common boundary segment, not just a point) needs more than four colors for each region to have a color not shared by any adjacent region.
- This is a horrendously difficult proof because it requires imagining, at least in theory, all possible types of maps that can be drawn. The first proof of the theory came in 1976, but it relied on computers in such a way that the proof could not be checked by humans. The authors of a simpler proof present their conclusion this way:
- "THEOREM 1. If T is a minimal counterexample to the Four Color Theorem, then no good configuration appears in T.
- "THEOREM 2. For every internally 6-connected triangulation T, some good configuration appears in T.
- "From the above two theorems it follows that no minimal counterexample exists, and so the Four Color Theorem is true. The first proof needs a computer. The second can be checked by hand in a few months, or, using a computer, it can be verified in about 20 minutes."
- In this case, the authors prove a negative ("There is no counter-example") which proves another negative ("There is no map ... which requires more than four colors").
Fermat's Last Theorem: x^n + y^n = z^n has no non-zero integer solutions for x, y and z when n > 2.
- The history of this theorem is quite funny. Fermat wrote the theorem in the margins of a book, then concluded: "I have discovered a truly remarkable proof which this margin is too small to contain." Indeed! Fermat died in 1665. His theorem was not proven for all values of n until 1995, and the proof certainly would not have fit in the margins of the book even if he had used every available square inch. Not only is this a proven negative, but it depends on proof of a negative: "Further work by other mathematicians showed that a counter-example to Fermat's Last Theorem would provide a counter-example to the Shimura-Taniyama-Weil Conjecture." By proving that the STW Conjecture was true--i.e., that there were no counter-examples--Andrew Weil proved that Fermat's Last Theorem was also true.
"All validating forms of categorical syllogism which have one negative premiss also have a negative conclusion."
Dr. Gary N. Curtis, Ph.D., majored in logic. His web site, "The Fallacy Files," is filled with fascinating and helpful material. He uses a fourfold classification of categorical propositions in some of his commentary on fallacies:
- General categorical sentences are deemed to be affirmative or negative in form, universal or particular in quantity ... So there are four kinds:
- A-type: affirmative universal; "every A is B"
- I-type: affirmative particular; "some A is B"
- E-type: negative universal; "no A is B"
- O-type: negative particular; "some A is not B"
- "Traditionally, the universal affirmative proposition was called 'the A proposition' and the particular affirmative was called 'the I proposition.' (The letters 'A' and 'I' come from the first two vowels in the Latin word, affirmo, 'I affirm.') The universal negative proposition was called 'the E proposition' and the particular negative was called 'the O proposition' (so called, from the vowels in the Latin word, nego, 'I deny').
Affirmative Negative Universal A: Every S is P. E: No S is P. Particular I: Some S is P. O: Some S is not P.
- Note the problem of quantification in ordinary English in the particular categories. "Some" means "one or more" and therefore the predicate should really be is/are. The universal affirmative could be written "All S are P."
- ". . . if either premise is negative, the conclusion must also be negative. For similar reasons, no affirmative conclusion about class inclusion can follow if either premise is a negative proposition about class exclusion. A violation results in the fallacy of drawing an affirmative conclusion from negative premises."
- There seem to be ten valid categorical syllogisms which lead to negative conclusions. These ten examples are quoted from "Establishing Validity." The names were given by medieval logicians. In each name, the vowels indicate the sequence of categoricals used in the argument. So, for example, "Baroco" has the sequence AOO.
A syllogism of the form AOO-2 was called "Baroco":
All P are M. Some S are not M. Therefore, Some S are not P.
The valid form OAO-3 ("Bocardo") is:
Some M are not P. All M are S. Therefore, Some S are not P.
Four of the fifteen valid argument forms use universal premises (only one of which is affirmative) to derive a universal negative conclusion: One of them is "Camenes" (AEE-4):
All P are M. No M are S. Therefore, No S are P.
Converting its minor premise leads to "Camestres" (AEE-2):
All P are M. No S are M. Therefore, No S are P.
Another pair begins with "Celarent" (EAE-1):
No M are P. All S are M. Therefore, No S are P.
Converting the major premise in this case yields "Cesare" (EAE-2):
No P are M. All S are M. Therefore, No S are P.
Begin with EIO-1 ("Ferio"):
No M are P. Some S are M. Therefore, Some S are not P.
Converting the major premise produces EIO-2 ("Festino"):
No P are M. Some S are M. Therefore, Some S are not P.
Next, converting the minor premise of this result yields EIO-4 ("Fresison"):
No P are M. Some M are S. Therefore, Some S are not P.
Finally, converting the major again leads to EIO-3 ("Ferison"):
No M are P. Some M are S. Therefore, Some S are not P.
"There is no such thing as a perfect voting system; every kind has one flaw or another" (Partha Dasgupta and Eric Maskin, "The Fairest Vote of All" [Scientific American, March 2004, 94).
The unstated condition in this quotation is that there are more than two candidates for a particular office. Dasgupta and Maskin argue for a system called "true majority rule" in which voters rank all of the candidates instead of just selecting one from the list.
Arrow's Impossibility Theorem:
"What Kenneth Arrow was able to prove mathematically is that there is no method for constructing social preferences from arbitrary individual preferences. In other words, there is no rule, majority voting or otherwise, for establishing social preferences from arbitrary individual preferences. This was a major result and for it and other work Kenneth Arrow received the Nobel prize in economics."
"It is not proven that George Mallory summited, however, it surely is not proven that he did not summit."
This is a fascinating combination of negatives. Both are examples of how negatives are determined in historical cases.
A. "It is not proven that George Mallory summited." The first kind of evidence needed to prove that Mallory reached the summit in 1924 would be direct observation by reliable witnesses. All of the potential witnesses testify that they did not see him on top of the mountain. The second kind of evidence would be photographic. Mallory may have carried a camera with him. His body has been found, but the camera has not been found. The third kind of evidence would be to find some artifact left by him on the summit. No complete search has been done yet; no artifacts have been reported being found by any successful expeditions; any artifacts he left may have been swept away since June of 1924. Hence the conclusion: there is no proof that Mallory reached the summit.
B. "It is not proven that he did not summit." "Mallory did not reach the summit" is the kind of negative that is extremely difficult to prove. We would need testimony from Mallory himself (who died on the mountain) or from reliable witnesses who watched him the whole time (but they were not, in fact, able to do so because of cloud cover) in order to establish that he did not reach the summit. One of Carl Sagan's favorite slogans is "Absence of evidence is not evidence of absence" (see [#absence below]). What we know for sure is that we don't know for sure how high Mallory climbed in 1924.
"George Mallory did not fall from the ridge of Mt Everest.". Mallory's body has been found in good condition. If he had fallen from the ridge, his body would be mangled, as are the bodies of all others who are known to have fallen from the ridge.
"The simultaneous measurement of two conjugate variables (such as the momentum and position or the energy and time for a moving particle) entails a limitation on the precision (standard deviation) of each measurement. Namely: the more precise the measurement of position, the more imprecise the measurement of momentum, and vice versa. In the most extreme case, absolute precision of one variable would entail absolute imprecision regarding the other."
This is Heisenberg's Uncertainty Principle. Physicists have no doubt whatsoever that it is true (if one grants the necessary definitions of its terms)--the principle does not mean that "everything in physics is uncertain."
"Pi cannot be written as any fraction."
"That is a matter of principle, unrelated to any issue of mere quantitative accuracy. Though the irrationality of pi was not proved until the late eighteenth century, the early Greeks did discover that numbers existed that could not be written as fractions, and this was both puzzling and shocking to them" (Stephen Hawking, God Created the Integers, xii).
"No mathematician has proven the existence of an odd perfect number."
"A perfect number is the sum of its integer divisors greater than or equal to 1 but less than itself, what are called its aliquot divisors. ... No one has ever discovered an odd perfect number. No mathematician believes that an odd perfect number exists [this is an unprovable negative assertion, since it is not possible to read the minds of all mathematicians]. But none has been able to prove that no odd perfect number exists" (Stephen Hawking, God Created the Integers, 5).
This particular negative has to be qualified in several ways. Perhaps there is a mathematician somewhere in the world who has come up with this proof but has not yet published it; or someone may falsify this statement in the future by coming up with a proof. The claim that "there is no proof of an odd perfect number" depends upon a series of ideas: if someone had found a proof, they would publish it; if it had been published, Hawking would have heard about it; if Hawking had heard about it, he would not have made the statement that he did.
"One simply cannot divide by zero."
The question--What number does one get when one divides the number two by zero?--admits no answer. The question is ill-formed. It is illegitimate" (William A. Dembski, Intelligent Design, 61).
"There is no general analytical solution for the three-body problem given by algebraic expressions and integrals."
"In 1887, mathematicians Ernst Bruns and Henri Poincaré showed that there is no general analytical solution for the three-body problem given by algebraic expressions and integrals. The motion of three bodies is generally non-repeating, except in special cases."
Don't Even Try to Contradict These Examples
If you can prove that I am wrong in any of my statements above, you will have proved a negative.
If you can't prove that any of my counterexamples are wrong, you have just admitted that I am right in thinking that negatives can be proven.
B. Examples of Unprovable Negatives
Note Well: I am not defending the proposition that "all negatives are provable" just that "some negatives are provable." This section does not really belong under "III. Proof by Counter-Example," but it is a handy place to collect the kinds of examples people give in support of the idea that "no negatives can be proven." They seem to think that a few examples can establish their thesis. The examples may be true but the universal negative conclusion is, I argue, false. "There are some negatives that cannot be proven" is a negative that can can be proven; that does not justify the conclusion that "no negative can be proven."
"There are no intelligent extra-terrestrial beings."
No human being has explored the whole of the physical universe. There could easily be a species or a multitude of species as intelligent as we are of which we have as yet found not trace. "The absence of evidence is not evidence of absence" say the folks involved in the Search for Extra-Terrestrial Intelligence (SETI; cf. Sagan, The Demon-Haunted World, 213).
"No aliens have ever visited earth."
I think there are multitude of good reasons to believe this, but I know it is not proven and that I do not have the resources to prove it decisively. To know this for sure, it would be necessary to have a complete record of the history of the earth in all places and all times over the last four billion years.
"This man will not harm the people committed to his care."
This is, of course, the great problem of the priest scandals in the Roman Catholic Church, but it is also common to making any appointment that carries authority with it (police officers, school administrators, teachers, Boy Scout leaders, day care staff, doctors, nurses, etc.). Since we cannot search the future, it is impossible to guarantee good behavior on the part of those who volunteer to serve. Both the innocent and the guilty say, "I am innocent. I will do no harm."
Jesus has been betrayed in every generation by those who have been selected to represent Him. I call this the Judas Factor because not even Jesus, true God and true man, was able to find twelve men who would remain faithful to Him in all circumstances. It would be lovely if we could be certain that "no candidates for priesthood will ever sin," but that gift has never been given to the Church. Sin happens. It is a shame and a scandal when it does, but it should not come as a surprise to Catholics when another priest is caught in sin. To guarantee a sin-free Church, God would have to turn us all (both the ordained and the unordained) into robots without free will.
I am persuaded of two negatives for which I cannot offer perfect proof:
1. God is not codependent. 2. God does not take hostages.
If God were to rob of us our freedom, He would take away our power to choose faith, hope, and love and therefore would rob us of our dignity as God-like persons. B.F. Skinner, an atheist, thought that freedom is an illusion and that we are, in fact, completely determined in our behavior by forces beyond our control. He advocated that we should accept our slavery to positive and negative reinforcement in his book, Beyond Freedom and Dignity. Against Skinner, the Church asserts a credible but not formally demonstrable negative: our behavior is not solely determined by social conditioning. We can and must take responsibility for our free choices. But where there is freedom, there is the possibility of sin. Therefore we cannot be certain that the people we ordain--or bless in marriage or appoint to office--will never sin against those who are in their care.
I'm sure there are lots of other examples of unprovable negatives. To draw the conclusion that "no negatives can be proven" from some examples of negatives that can't be proven is an example of a hasty generalization ("Also Known as: Fallacy of Insufficient Statistics, Fallacy of Insufficient Sample, Leaping to A Conclusion, Hasty Induction"
"This fallacy is committed when a person draws a conclusion about a population based on a sample that is not large enough. It has the following form:
"Sample S, which is too small, is taken from population P. Conclusion C is drawn about Population P based on S.
"The person committing the fallacy is misusing the following type of reasoning, which is known variously as Inductive Generalization, Generalization, and Statistical Generalization:
"X% of all observed A's are Bs. Therefore X% of all A's are Bs.
"The fallacy is committed when not enough A's are observed to warrant the conclusion. If enough A's are observed then the reasoning is not fallacious."
IV. Let the Equivocation Begin
A. "You have completely misrepresented my position. What I meant when I said 'you can't prove a negative' is that 'you can't prove a negative without proving something positive first.'"
My mistake. That means there is no disagreement between us at all, because you have adopted my position.
There is a double negative in the new proposition to which you have just shifted your allegiance. By the rules of formal logic, I may rewrite your statement positively to eliminate the double negative: "You can only prove a negative by proving something positive first." In saying this, you have conceded that negatives can be proven; now you are only fussing about the method of proof.
But this new horse is not the one you rode into mid-stream. These two propositions are not logically the same:
"No one can prove a negative."
"No one can prove a negative except by proving a positive."
If they were identical, I could write them this way:
"No one can prove a negative" = "No one can prove a negative except by proving a positive."
Now I may substitute for the double negative:
"No one can prove a negative" = "One can prove a negative only by proving a positive."
And, since you've told me that the trailing phrase really doesn't change the meaning of the proposition, I feel confident that you won't mind me removing the trailing phrase from the last line. If you do object, you will have to defend the idea that no one can prove a positive, which will be a bit of a stretch, I think.
"No one one can prove a negative" = "One can prove a negative."
It seems like a funny way of talking, but if that's how you think about things, I guess it's OK with me. I'm not going to hire you to tutor me in the scientific method, however.
B. "You've misrepresented my position again. What I mean is that it is trivial to prove a negative from a positive."
I guess this argument goes like this:
The first element in a proof is what is really proven.
Other elements in a proof depend on what was really proven.
Therefore other elements in the proof are not really proven because they are dependent truths.
Well, that's a very emotional way of looking at things. Were you a middle child? Do you think the first-born children got all the good stuff and you only had leftovers and hand-me-downs?
In the fields of logic and mathematics, there are no varieties of truth. If a proposition is derived from another proposition using the rules of inference, it does not matter that it comes second.
And, as in the first case, you are again conceding that negatives can, in fact, be proven. You just object to the fact that some proofs require several steps.
When a lawyer presents an alibi for the client in court, the argument runs like this:
|Premise: Human beings can't be in two different places at the same time.||This is something we take for granted. It is based on the experience of being human.|
|Assertion: My client is a human being.||The jury can see this for themselves.|
|Conclusion: My client can't be in two places at the same time.||Modus ponens. If p implies q, and p is true, then q follows.|
|Premise: Human beings are incapable of using telekinesis to injure each other.||Another fact about human nature that is generally not argued in court.|
|Assertion: My client is a human being.||Direct observation.|
|Conclusion: My client could not have committed the crime by using telekinesis.||Modus ponens.|
|Fact: The crime took place at such-and-such a time in such-and-such a place.||That people can know such facts is one of the essential elements of our criminal justice system. We can argue elsewhere about the epistemology of testimony.|
|Fact: My client was elsewhere at the time of the crime.||This is a positive statement that can be proven, within the rules of proof for our legal system, by the use of evidence (some form of human testimony).|
|Conclusion: My client could not have committed the crime.||
This is the desired and essential negative that the lawyer is striving to prove. It depends on the previous theorems that "no human being can be in two different places at the same time" and "humans cannot injure each other using telekinesis."
The definition of "different places" will vary with the type of crime committed and the nature of the exonerating evidence that the lawyer can produce. It may be sufficient to establish that the client was in a different room from where the crime took place.
It is not fair to object that the lawyer uses a "because" statement to demonstrate the client's innocence. If the evidence is good that the accused was elsewhere, we can conclude that the person did not commit the crime because of that absence from the crime scene.
Courts take the two facts about human nature for granted. I doubt very much that any judge is going to allow the hypotheses of bilocation or telekinesis to be used by a prosecutor to obtain a conviction.
Yes, of course, I've read dozens of murder mysteries in which the person with the best alibi turns out to be the murderer. That's what makes fiction fun. Murderers do figure out how to give the illusion of being elsewhere or else they work out a method of murder that does not require their presence. The best novels are persuasive about the effectiveness of the techniques and do not resort to the use of the malicious use of superpowers to achieve their effect. The most recent novel I read with a very clever false alibi was Have His Carcass by Dorothy Sayers. She helped to found the Detection Club, part of whose initiation ritual reads:
"Do you promise that your detectives shall well and truly detect the crimes presented to them, using those wits which it may please you to bestow upon them and not placing reliance on nor making use of Divine Revelation, Feminine Intuition, Mumbo-Jumbo, Jiggery-Pokery, Coincidence or the Act of God?"
C. "You only picked easy examples of 'proven negatives.'"
Somewhat true. I would like the argument to be accessible to anybody who cares to think critically about what is wrong with the maxim that there are no provable negatives.
I think the possibility of proving negatives could be expressed in some kind of formula that would correlate the size of the search space and the cost of searching such a space. Searching my room for elephants is pretty easy and it doesn't cost me much. Searching the entire universe from the beginning to the end of time to prove the non-existence of extra-terrestrial intelligence is beyond my budget. Examining all possible negatives in detail is impossible even for people richer than I am, because it is either an infinite set or at the very least such a large set that a whole lifetime of searching would not make a dent in the total. Mathematicians can deal with infinite sets, as in Fermat's Last Theorem, or with huge sets such as in the Four Color Theorem, because they can get at the characteristics of every member of the set without looking at each one of the elements in detail.
D. "You are such a dolt. Nobody ever proves anything in science."
MX: "Astronomers have proven that the earth orbits the sun, not vice-versa."
RR: "This is not the correct way to think about the problem (and I should know, being a professional astronomer)."
MX: "Your qualifications as an astronomer may or may not be helpful in settling a philsophical debate. The argument from authorith is the weakest form of argument; the argument from one's own personal authority is weaker still."
RR: "You're not proving that 'the Sun does not go around the Earth'. What you're doing is comparing two hypotheses:
1) The Sun goes around the Earth.
2) The Earth goes around the Sun."
MX: "Calling them 'hypotheses' doesn't change the logical relationship between them. If the first proposition is true, the second is false, and vice versa."
RR: "Once you have your hypotheses, then you gather evidence (in this case astronomical observations), and determine which hypothesis fits the facts most closely. That hypothesis then becomes the theory of how the Earth and Sun interact, and will remain the leading theory until another hypothesis comes along to challenge it. Then, the cycle starts again."
MX: "Your contention that 'There are no proofs in astronomy, only theories' is not an astronomical observation, but a philosophical assertion. It should be embraced with no more enthusiasm than the evidence warrants. You cannot verify this assertion by the use of any astronomical instruments (telescope, interferometer, gravitometer, chronometer, etc.). That is why your qualifications as an astronomer are irrelevant. Your assertion is a meta-astronomical proposition because it is metaphysical. I argue that the standard-issue definition of the terms used in deciding the question of heliocentrism vs. geocentrism--gravity, mass, dimension, cause, motion, momentum, stellar parallax, orbit--are sufficent to answer the question of which body orbits the other without any residue of doubt. I know that theories of gravity and mass are incomplete, but I deny that astronomers will some day discover that the earth is bigger than the sun and holds the sun in orbit around itself by means of the force of gravity. The proposition that the earth orbits the sun, and not vice-versa, will not be changed by any conceivable revision in astronomy, because I am using the present-day meaning of the words."
RR: "If the two answers both can fit, but one takes a lot more 'shoehorning' to do so, then the simplest one is the one chosen. In the case of the Sun/Earth problem, the idea of epicycles was used to fit the orbital motions of the planets; while epicycles worked very well, it was extremely difficult to figure out the proper values. Kepler's theories of planetary motion are easy in comparison, so they became the viable explanation for what we see in the sky."
MX: "This is a red herring, but I'll chase it anyway. It has nothing to do with proving or disproving negatives. Your account of the history of science is incomplete and, I think, misleading. Please note that you cannot study history with a telescope. Your qualifications as an astronomer allow you to work in an observatory or to participate in the interpretation of data, but they do not qualify you to read history off the top of your head. The Ptolemaic theory had no 'reason' for the planets to travel in cycles and epicycles, except for the Platonic assumption that the motions of the heavens must be perfect, which meant pure circular motion at unchanging velocities. Kepler also had no reason to explain his laws of motion. They were derived just from the facts, and were a triumph of hard work and genius (an extraordinary ability to imagine or visualize physical relationships). When Newton integrated Galileo's laws of motion with his own law of gravity, then Kepler's system emerged as a function of the force of gravity acting on bodies in motion. There is far more complexity in the integrated understanding than your comment suggests. Your claim that scientists chose the simpler system is simplistic and false. Kepler had to invent new forms of mathematics to derive the elliptical orbits from his data. Newton invented a version of calculus to allow him to deal with smoothly changing velocities. The Ptolemaic system had its difficulties, but it was far simpler than Newtonian orbital dynamics."
RR: "This is all just how the scientific method works... as a professional astronomer, I'm amused sometimes reading this forum at how little people seem to understand how to apply this method."
MX: "Another red herring. I love distractions. Here we go again: you can't observe the 'scientific method' by the use of the tools of your trade. It is not out in the starry heavens nor in cold dark matter nor hidden in a black hole. When you start expressing opinions about 'the scientific method' and how it applies in other fields, you have ceased to act as an astronomer and have begun to do philosophy. I don't mind that. I like philosophy. I guess that makes me a philophilosopher or a philosophile. If the debates on the forum could be resolved by the use of the instruments of astronomy, I would bow to your authority and take you at your word. You would simply point the Hubble Telescope in our direction and announce the results of your observation, while noting the limits of precision of the instrument. But you cannot use astronomical methods to solve an archeological problem. It's nice to know your profession, but it's pretty irrelevant to the set of questions under consideration in the forum."
RR: "... In other words, proving [the TIGHAR hypothesis] wrong wouldn't be proving a negative, but it would just show that another answer is the correct one. ..."
MX: "OK, we've come back to the basic question. The TIGHAR hypothesis is that Amelia Earhart and Fred Noonon landed on Gardner Island, now known as Nikumaroro or Niku, on July 2, 1937. If the Niku hypothesis is proven wrong, that means that AE and FN did not land on Gardner Island. If that's not a negative, what is?"
E. "Well, that's not what I mean by a negative."
Tell me what you mean by a negative, and I'll either show you that there are provable negatives that meet your definition or else I'll show you that your definition is equivocal.
I have given quite a range of what I think of as "negatives." Perhaps they can be classified:
Some negatives take the form "There is/are no ..."
There are no proven negatives.
There is no God.
There is no Santa Claus.
There is no Easter Bunny.
There is no tooth fairy.
There was no rear antenna mast on the bottom of Amelia Earhart's plane when she took off from Lae on July 2, 1937.
There is no rule in logic against proving negatives.
There are no weapons of mass destruction in Iraq.
There is no atmosphere on the moon.
There is no weather on the moon.
There is no erosion caused by wind and weather on the moon. Therefore, collecting moon rocks of a certain kind gives an excellent view of the moon's earliest state.
These and other propositions expressing the same idea, though using different words, are negative existentials. Bertrand Russel, the dude who took 362 pages to prove that 1 + 1 = 2 and who knew a thing or two about formal logic, showed how true negative existential beliefs could be represented and defended in his system of the propositional calculus.
Some negatives are simply marked by the word "not" modifying the predicate:
The Eiffel Tower is not in New York City.
The Twin Towers are not in New York City.
I am not sixty years old (this will only be true until August 21, 2012--if I live that long).
This essay is not written in pen-and-ink.
My computer does not understand this sentence as I do.
Some negatives are formed with adjectives modifying the subject or words which emobody a negation:
No one can fly to the moon using a human-powered machine.
Nobody knows anything for certain.
Nemo dat quod non habet ("No one can give what they do not have.")
Everyone who thinks they have been abducted by aliens is deluded.
No two electrons in an atom can have identical quantum numbers (Pauli's Exclusion Principle).
Some negatives deny possibilities:
You cannot breathe 100% nitrogen and survive for more than a couple of minutes at most..
You cannot cross a ten-foot chasm in three easy steps.
You cannot turn lead into gold by breaking and/or forming any series of chemical bonds.
The strong force that holds protons together will not hold electrons in the nucleus of the atom.
F. "That's not what I mean by proof."
Same challenge. Tell me what you mean by proof, and I'll either meet your standards or show you that the standards contain an inconsistency. The maxim suggests that there is something rather important about "proof," otherwise it wouldn't be lamentable that "negatives can't be proven."
Here is a pretty usable definition of proof. I have highlighted and questioned one phrase that I don't know whether I should accept or not. Other than that, it's a pretty good characterization of what it means to prove something in a formal system:
"A proof is any sequence of formulas, each of which is an axiom or an immediate consequence, by a rule of inference, of (one or two) preceding formulas of the sequence. Any proof is said to prove its last formula, which is called a theorem or provable formula of first-order intuitionistic predicate logic [???]. A derivation of a formula E from a collection F of assumptions is any sequence of formulas, each of which belongs to F or is an axiom or an immediate consequence, by a rule of inference, of preceding formulas of the sequence, such that E is the last formula of the sequence. If such a derivation exists, we say E is derivable from F."
For questions that are not matters of formal logic, what I mean by "proving them true" varies with the particular kind of thing involved in the question. For sensible objects (keys, coins, cars, giraffes), the proper use of the senses will prove or disprove a statement about the sensible reality. For theoretical constructs (orbits, mass, energy, wavelengths, fields, spectra) our senses and reasoning have to be extended by the use of instruments and conceptual frameworks.
"What skeptical thinking boils down to is the means to construct, and to understand, a reasoned argument and--especially important--to recognize a fallacious or fraudulent argument. The question is not whether we like the conclusion that emerges out of a train of reasoning, but whether the conclusion follows from the premise or starting point and whether that premise is true."
Let's run this debate through the whole of Carl Sagan's "Baloney Detection Kit." All page references are to the first edition of The Demon Haunted World.
Sagan's Positive Precepts
1. "Wherever possible there must be independent confirmation of the facts" (210).
Where I have made my own argument, the confirmation is in the logic itself.
Where I have referred to facts established by others, I have given references.
You may check my work by checking my argument or by checking my sources.
2. "Encourage substantive debate on the evidence by knowledgeable proponents of all points of view" (210).
I have created this page precisely to encourage debate on this topic.
3. "Spin more than one hypothesis" (210).
I have. I have several lines of argument: from self-referential inconsistency; from the fact that some negatives can be proven; from the absence of any proof of the maxim in question; from the insight that not all negatives can be tested.
"Retrospective studies show that some jurors make up their minds very early--perhaps during opening arguments--and then retain the evidence that seems to support their initial impressions and reject the contrary evidence. The method of alternative working hypothesis is not running in their heads" (210, footnote to this tool).
I'm open to persuasion that I've made a mistake in thinking that some negatives can be proven and that the maxim is true. But I won't change my mind unless I am presented with reasoned arguments and evidence that establish its truth in the form "No negatives can be proven."
4. "Try not to get overly attached to a hypothesis because it's yours" (210).
This essay, such as it is, is mine. I am responsible for each sentence in it that is not attributed to some other source. I am unquestionably "attached" to it because I have invested a great deal of time and energy in putting it together. But I don't think the essential hypothesis ("Some negatives can be and have been proved") belongs to me in any way, shape, or form. This essential thesis belongs to anyone who cares to think about thinking in a responsible fashion. I believe it existed before I came into being and that it is utterly independent of anything I have written here.
5. "Quantify" (211).
The quantities involved in this argument are 'all' and 'some'. The issue is between "All negatives are unprovable" and "Some negatives are provable."
6. "If there's a chain of argument, every link in the chain must work (including the premise)--not just most of them."
I have essentially three distinct and independent arguments against the maxim:
A. It is nonsensical considered in itself.
B. The facts are against it:
Examples I have made up.
Examples from various sciences.
C. It is a claim that has not been and (I argue) cannot be proven.
Any one of these "chains of reasoning" is sufficient to defeat the argument. They do not form a chain as a group, though they do arrive at the same conclusion.
7. "Occam's Razor: This convenient rule-of-thumb urges us when faced with two hypotheses that explain the data equally well to choose the simpler" (211).
I don't think this rule applies in this case. I don't think either hypothesis is simpler than the other. The generalization from a few unprovable negatives is simplistic, but it is not simpler than thinking "Some negatives can be proved."
8. "Always ask whether the hypothesis can be, at least in principle, falsified" (211).
The argument I am making is that "some negatives can be proved." The meaning of the argument is that this thesis cannot be falsified. I don't know what it would mean to think that it could be falsified "at least in principle." I have done my level best to consider what thought would be like if it were false and I conclude that it cannot be.
9. "Control experiments are essential" (211).
The proper laboratory for testing metaphysical maxims is in the mind. There is no other place where "controlled experiments" can be run on this type of assertion.
10. "Variables must be separated" (211).
There aren't a lot of variables in this argument. If there is one proven negative, the maxim that "you cannot prove a negative" is wrong and the contrary idea that "some negatives are provable" is true.
Errors to be Avoided
1. "Ad hominem--...attacking the arguer and not the argument" (212).
I do not argue that the maxim is false because the people who propose it are poorly educated or lack the skills of philosophical investigation (though I do think that is the case). I argue the maxim is false because it fails to meet the standards of logic and of factuality.
2. "Argument from authority" (212).
As I have indicated elsewhere in this brief, I believe there is a place for authority in science. I do not have the skills to personally check all of the demonstrations of negatives that I have adduced as evidence. It would be foolish for me to say that the theorems of mathematics are not true because I have not personally checked them all.
I have great respect for people who know more than I do. If someone says, "I am a lawyer," I will listen with respect to what they tell me about law. If someone says, "I am an astronomer," I will let them teach me about the structure of the universe as known through astronomical observation. If someone says, "I am a board-certified forensic image investigator," I will accept what they tell me about the interpretation of images. It seems to me unreasonable to think that I am as qualified as these experts are in their chosen fields.
3. "Argument from adverse consequences" (212).
I have not used this argument to make my case. Because I think the case is valid, I do believe that those who advocate the maxim are teaching a falsehood and I do find that lamentable. But I do not rest my case on this premise.
4. "Appeal to ignorance--the claim that whatever has not been proved false must be true and vice versa" (213).
I will quote the entire paragraph under this heading because it gives two more examples of unprovable negatives and concludes with a famous Saganism:
"Appeal to ignorance--the claim that whatever has not been proved false must be true and vice versa (e.g., There is no compelling evidence that UFOs are not visiting the Earth; therefore UFOs exist--and there is intelligent life elsewhere in the Universe. Or: There may be seventy kazillion other worlds, but not one is known to have the moral advancement of Earth, so we're still central to the Universe.) This impatience with ambiguity can be criticized in the phrase: absence of evidence is not evidence of absence." (213)
I certainly do not violate this principle in these reflections. Quite the contrary. I believe that those who chant the maxim that "no one can prove a negative" are operating on this principle.
5. "Special pleading" (213). Sagan does not define this fallacy. He just gives examples of it. Here is a good description:
"Description of Special Pleading Special Pleading is a fallacy in which a person applies standards, principles, rules, etc. to others while taking herself (or those she has a special interest in) to be exempt, without providing adequate justification for the exemption.
"This sort of "reasoning" has the following form:
1. Person A accepts standard(s) S and applies them to others
in circumtance(s) C.
2. Person A is in circumstance(s) C.
3. Therefore A is exempt from S.
"The person committing Special Pleading is claiming that he is exempt from certain principles or standards yet he provides no good reason for his exemption. That this sort of reasoning is fallacious is shown by the following extreme example:
1. Barbara accepts that all murderers should be punished for their crimes.
2. Although she murdered Bill, Barbara claims she is an exception because she really would not like going to prison.
3. Therefore, the standard of punishing murderers should not be applied to her.
"This is obviously a blatant case of special pleading. Since no one likes going to prison, this cannot justify the claim that Barbara alone should be exempt from punishment.
"From a philosophic standpoint, the fallacy of Special Pleading is violating a well accepted principle, namely the Principle of Relevant Difference. According to this principle, two people can be treated differently if and only if there is a relevant difference between them. This principle is a reasonable one. After all, it would not be particularly rational to treat two people differently when there is no relevant difference between them. As an extreme case, it would be very odd for a parent to insist on making one child wear size 5 shoes and the other wear size 7 shoes when the children are both size 5."
I have made no special pleas in this brief. I submit to the rules of logic and evidence just as I expect others to do.
6. "Begging the question, also called assuming the answer" (213).
The arguments I have advanced do not assume their own conclusions except insofar as the first principles of thought are employed without being proved. No one can construct an argument without using those first principles, so I feel that I am in good company in using them without proving them.
7. "Observational selection, also called the enumeration of favorable circumstances, or as the philosopher Francis Bacon described it, counting the hits and forgetting the misses" (213-14).
This seems to me to be the problem with the advocates of the maxim. They take a few juicy examples and use it to 'justify' a false generalization.
I am allowed to select a few examples because the maxim is proven false if even one example I give is true. I am not defending the idea that "all negatives are provable." That is a straw man (see below). The hypothesis that "No negative is unprovable" is not sustained by a finite set of examples that "some negatives are unprovable."
8. "Statistics of small numbers" (214).
This is not a statistical argument. I don't object to people saying "Most negatives are unprovable" or "Many negatives are unprovable." It's a respectable opinion and quite different from the proposition that "No negative hypothesis can be proven." Of course, to be academically respectable, one should take a representative sample of negatives and demonstrate that the word 'most' is appropriate, but I won't demand such rigor from folks who just want to air out an opinion.
9. "Misunderstanding the nature of statistics" (214).
Sagan just makes a joke here: "[A president he dislikes] expressed astonishment and alarm on discovering that fully half of all Americans have below average intelligence." Sagan gives no evidence that the president in question ever committed this fallacy. The joke has a thousand variations.
10. "Inconsistency" (214).
Sagan again gives no definition, just examples. I've tried to be consistent in my use of terms. I would be happy to revise this essay (again) if someone can point out my inconsistencies.
11. "Non sequitur--Latin for 'it doesn't follow'" (214).
This is the claim I make against the maxim: from the examples given of unprovable negatives, it does not follow that "no negative hypothesis can be proven."
12. "Post hoc, ergo propter hoc"--Latin for, 'It happened after, so it was caused by'" (215).
There is no causal or temporal analysis involved in this debate.
13. "Meaningless question" (215).
If the suggestion that "No negative hypothesis can be proven" is a meaningful statement, then the contrary view that "Some negative hypotheses can be proven" is equally meaningful.
14. "Excluded middle, or false dichotomy--considering only the two extremes in a continuum of intermediate possibilities" (215).
This is a poor choice of names for this fallacy. Where there is no continuum and a statement is being compared to its opposite, the Principle of Excluded Middle is just an ordinary feature of orthodox logic. It would be a false dichotomy to say that we have only two choices. "Either all negatives are provable or none are." I defend the third alternative: some are provable and some are not.
15. "Short-term vs. long term" (215).
This argument has no time-dimension to it.
16. "Slippery slope" (215).
I do believe that uncritical thinking bears bad fruit, but I do not base my argument against the maxim on this belief.
17. "Confusion of correlation and causation" (215).
As I said in point 12, there is no causal analysis involved in this debate.
18. "Straw man--caricaturing a position to make it easier to attack" (215).
This is a sloppy definition of what a "straw man" is. It is not just a caricature (emphasizing some features at the expense of others) but a complete misrepresentation. If someone attributes the wrong hypothesis to me ("Moleski thinks all negatives are provable"), then produces one example of an unprovable negative as a counterexample, they have not defeated my argument but just the "straw man" or "paper tiger" that they set up.
19. "Suppressed evidence, or half-truths" (216).
So far as I can tell, I haven't held back any part of the argument that is relevant to deciding between the alternative hypotheses. If I or my respondents can add anything to the debate, I will burn a few more recycled electrons to add it to this page.
20. "Weasel words" (216). No definition given.
This is what I call "equivocation." There may be other variations that fall into this same category. It's not fair to change the meaning of words in the middle of a debate, so that the meaning of the words in the conclusion is different from the meaning in the statement of the question or the premisses. I've done the best I know how not to change the meaning of the terms I use. I expect the same courtesy from others.
One example of "weasel words" is to make a distinction without a difference. For the purposes of this essay, I haven't made a distinction between thesis, hypothesis, thesis, theory, proposition, or premise. I use these words pretty much interchangeably for the sake of "elegant variation." If push comes to shove, I'll make up abbreviations for the statement that I reject and for the alternative that I defend, and edit the whole darned page so that I never vary in my terminology. I hate reading that kind of writing, but I will adopt that style if necessary.
G. "Oh, it's all just semantics."
Even scientists rely on semantics. They strive, as best they can, to define their terms and stick with the defined meaning through the course of a debate.
If you wish to withdraw the proposition because you don't know what you mean by it, that's fine with me. "I don't know whether negatives can be proven" is a perfectly respectable position to take, as is "I don't know how to use words consistently when I am forming an argument."
H. "But everybody says it. Why shouldn't I?"
The argument from authority is the weakest form of argument, according to Aristotle (in appealing to the authority of Aristotle to repudiate appeals to authority, I am, of course, being self-referentially inconsistent!). The argument from "science says" or "all reputable authorities agree" or "everybody thinks" (argumentum ad populum) is the weakest form of the argument from authority. Unless you are gifted with ESP, the only way to determine what "science says" or "all authorities think" or what "everybody thinks" is to do a survey using the instruments of sociology.
I predict that the results of such a survey, if conducted fairly, would show a diversity of opinion. It is an unassailable truth that "people disagree." If you disagree with this proposition, you prove my point. My impression is that, by far, the vast majority of people who invoke this principle do so as a matter of "cultural conditioning." They are adopting a slogan without thinking critically about what it means and what it would take to establish it or to show that it is untenable. I don't mind adopting a minority position in this and in other cases. I do not subscribe to the view that truth and falsehood are to be determined by taking public opinion polls. I am accustomed to the fact that many people disagree with my view of reality.
I am not alone in serving ideals that the multitude do not understand or embrace:
- "The ideal mathematician's work is intelligible only to a small group of specialists, numbering a few dozen or at most a few hundred. This group has existed only for a few decades, and there is every possibility that it may become extinct in a few decades. However, the mathematician regards his work as part of the very structure of the world, containing truths which are valid forever, from the beginning of time, even in the most remote corner of the universe. He rests his faith on rigorous proof; he believes that the difference between a correct proof and an incorrect one is an unmistakable and decisive difference. He can think of no condemnation more damning than to say of a student, 'He doesn't even know what a proof is.'
- "Yet he is able to give no coherent explanation of what is meant by rigor, or what is required to make a proof rigorous. In his own work, the line between complete and incomplete proof is always somewhat fuzzy, and often controversial."
- "His writing follows an unbreakable convention: to conceal any sign that the author or the intended reader is a human being. It gives the impression that, from the stated definitions, the desired results follow infallibly by a purely mechanical procedure. In fact, no computing machine has ever been built that could accept his definitions as inputs. To read his proofs, one must be privy to a whole subculture of motivations, standard arguments and examples, habits of thought and agreed-upon modes of reasoning. The intended readers (all twelve of them) can decode the formal presentation, detect the new idea hidden in lemma 4, ignore the routine and uninteresting calculations of lemmas 1,2,3,5,6,7, and see what the author is doing and why he does it. But for the noninitiate, this is a cipher that will never yield its secret. If (heaven forbid) the fraternity of non-Riemannian hypersquarers should ever die out, our hero's writings would become less translatable than those of the Maya."
- ". . . we must pause to realize that, outside our coterie, much of what we do is incomprehensible. There is no way we could convince a self-confident skeptic that the things we are talking about make sense, let alone 'exist.'"
- "For non-math types resistance may be the honest reaction to innate limitations. Not everyone becomes a piano player or an ice skater. Why should it be otherwise for mathematics?"
I think the same principle holds true for logic and philosophy: not everyone is equally adept at reasoning about reasoning (cf. Bernard J. F. Lonergan, Insight).
I. "I have the right to my opinion."
True. Then in the future, it would be good for you to say, "I believe that no negatives can be proven." This will indicate clearly that it is a religious act and not a conclusion from the field of logic, mathematics or science.
J. "I just think you're wrong."
I never said I could change your mind. I can't prove anything to you without your permission and cooperation. It is an unassailable truth that I might be wrong. If you proved me wrong in making this statement, you would show that it was a sound judgment. I am not infallible. I have made mistakes in the past and I will probably make many more if I live long enough and funds hold out. But I do wish you wouldn't call what you're doing "thinking." It gives thought a bad name. Could I get you to say instead, "I just feel you're wrong"? Intuition is not all bad. There is a place in science for hunches, guesses and feelings. When your deep-seated passion has led you to find evidence, please let me know what it is. I love testing hypotheses. It makes me feel good all over.
K. "You haven't proven me wrong."
The burden of proof falls on the person who makes the claim. "No negative hypothesis can be proved" is a hypothesis in need of proof. It ought not to be given any more weight than the evidence for it allows. Theories, unlike persons, are not innocent until proven guilty. They are unusable as theorems until they are proven to be true. I've done my level best to show why I reject the hypothesis as untenable, both in terms of abstract logic and in terms of matters of fact.
L. "What I meant was that it is hard to prove a negative."
That's true. I have no objection to people saying that. But that is a different proposition from the maxim in question.
M. "Your brief is too long. I can't read the whole thing."
Sorry about that. I have trouble reading the whole thing, too. It wasn't this long or complicated when I got started on it. The nature of proof is a topic that has intrigued me since I took Thomas Aquinas as my confirmation sponsor in the spring of 1965, and my interest in "theological epistemology" (a term popularized, if not coined, by Avery Dulles, SJ) has been the foundation of my work in philosophy and theology. I think that explains why this note has grown so complex since the first draft in 2003.
There is a debate in formal logic about the shortest possible proofs; there is no rule about how long a proof may be. Mathematicians criticize the proof of the Four Color Theorem on aesthetic grounds, but that does not change the conviction that proof has been offered:
- "When I heard that the four-color theorem had been proved, my first reaction was, 'Wonderful! How did they do it?' I expected some brilliant new insight, a proof which had in its kernel an idea whose beauty would transform my day. But when I received the answer, 'They did it by breaking it down into thousands of cases, and then running them all on the computer, one after the other,' I felt disheartened. My reaction then was, "So it just goes to show, it wasn't a good problem after all.'"
I've tried shorter versions of this argument in e-mail and other conversations with people. The short exchanges haven't proven satisfactory. Hence this extended reflection.
N. "You're a crackpot."
That well may be true. I have more evidence of the truth of this assertion than I care to adduce here. But this is just an ad hominem remark and not a refutation of the arguments I have presented above. If you can show me that I've made an error in reasoning, I will change my mind and publish the results here. If you just don't want to talk to me any more, that's your choice. I've had fun putting this together. I enjoy thinking about thinking. I wish you well in your own quest to become more rational.
V. Concluding Uncharitable Postscript
- January 28, 2006 -- Feast of St. Thomas Aquinas
I've just finished reading Simon Singh's account of how Andrew Wiles proved Fermat's Last Theorem (Fermat's Enigma). Wiles integrated three centuries of developments in mathematics to prove this great negative. I wondered why anyone who claimed to be a thoughtful person would say, "You can't prove a negative." I hazard a guess: the motivation to make this claim comes from the existence-of-God debates. I conjecture that, after being confronted so many times by the claim that no one can prove that God does not exist, someone replied, "Our inability to prove that there is no God is just one instance of the general rule that you can't prove a negative."
Carl Sagan firmly believed that there is extra-terrestrial intelligent life in the universe. From his materialistic assumptions it follows that there must be life that is at least as intelligent as we are elsewhere. Since life happens here merely by accident, it must happen elsewhere in the universe, too. The laws of physics, chemistry, and (materialistic) evolution are the same everywhere in the universe. There is nothing unique about the earth or about humanity (an unproven negative that makes good sense to me--all of our astrophysics is based on the assumption that the earth is a representative sample of the whole cosmos). The same kind of things that happen here by the luck of the draw must happen elsewhere by the luck of the draw. And no one can prove that there is no extra-terrestrial intelligent life in the universe. It's too big a space for us to search. We don't even know for sure how big the search space is--all we know is that the horizon we can "see" is limited by the distance that light can have traveled since the Big Bang (which is directly proportional to the length of time that has elapsed since the Big Bang--something like 15 billion years--and the rate of expansion of the universe since then).
So Sagan and I are in the same boat. I believe in God and I am confident that my opponents cannot prove His non-existence. Sagan believed in extra-terrestrial intelligence and he was confident that his opponents could not prove their non-existence. I imagine that believers in UFOs and alien abduction are equally confident that no one can disprove that we have been visited and probed by aliens. The ground of our confidence is not the imaginary principle that "no negative can be proven" but the fact that we cannot conduct the kind of search necessary to affirm the negative with certainty.
- Carl Sagan, The Demon-Haunted World: Science as a Candle in the Dark. New York: Random House, 1996, 28. Hereafter Sagan (1996).
- Sagan (1996), 171.
- Sagan (1996), 210.
- Sagan (1996), 315.
- Sagan (1996), 254.
- Sagan (1996), 259.
- Philip J. Davis and Reuben Hersh, The Mathematical Experience with Introduction by Gian-Carlo Rota (Boston: Houghton Mifflin, 1981), 150. Hereafter D&H.
- (D&H, 18).
- (D&H, 20-21).
- The page from which this quotation was taken (http://gncurtis.home.texas.net/manyques.html) no longer exists. So much for my authority! I'll have to dig up another, I guess.
- (D&H, 217).
- (D&H, 340).
- ("Portraits from Memory"; D&H, 333).
- (D&H, 334).
- Sagan (1996), 66.
- Sagan (1996), 69.
- "Magnetic Materials."
- How Everything Works, Question 229.
- "How Everything Works," Question 1233.
- "How Everything Works," Question 807.
- "Diamagnetic, Paramagnetic, and Ferromagnetic Materials."
- "Physics News."
- "Monopole FAQ."
- "Squaring the Circle."
- "Ask Dr. Math FAQ: Squaring (Quadrature of) the circle."
- "Ask Dr. Math FAQ: Doubling (or Duplicating) the cube."
- BBC News.
- Neil Robertson, Daniel P. Sanders, Paul Seymour and Robin Thomas, "Four Color Theorem."
- "Fermat's Last Theorem."
- Introduction to Logic (10th Edition, 1998), Copi and Cohen, pp. 277-8.
- "Logic 101."
- "Establishing Validity."
- "Arrow's Impossibility Theorem for Aggregating Individual Preferences into Social Preferences."
- "Mallory and Irvine, The Final Chapter: Our Theory ... (10/2004)."
- "Everest: Mallory and Irvine The Final Chapter: Q&A January 2005 Part Two."
- Wikipedia, "Three-Body Problem."
- The Nizkor Project, "Fallacy: Hasty Generalization.")
- The Nizkor Project, "Fallacy: Hasty Generalization."
- "The Secret Life Of Eric The Skull: Dorothy L. Sayers And The Detection Club."
- "Intentionality: True negative existential beliefs."
- "Intuitionistic Logic."
- Sagan (1996), 210.
- The Nizkor Project, "Fallacy: Special Pleading.".
- (D&H, 34).
- (D&H, 36-7).
- (D&H, 44).
- (D&H, 283).
- (D&H, 384).
- Carl Sagan, The Demon-Haunted World: Science as a Candle in the Dark. New York: Random House, 1996.
- Philip J. Davis and Reuben Hersh. The Mathematical Experience. Introduction by Gian-Carlo Rota. Boston: Houghton Mifflin, 1981.
- Informal logic. Includes a list of links to other sites.
- The Fallacy Files by Dr. Gary N. Curtis, Ph.D.
- Carl Sagan's "Baloney Detector Kit."
- Wikipedia, "You Can't Prove a Negative."
- Hales, Steven D. (2005). "Thinking Tools: You can Prove a Negative" (Think 4 (4): 109–112).
First edition: 26 August 2003.
I developed this essay in HTML on my College website. It was last updated there on 20 February 2012. On 23 June 2013, I think I've got the whole thing into decent shape for the wiki. Subsequent revisions can be traced by examining the history of this page.
Comments and Corrections Welcome
Please e-mail me if you have spotted something that needs to be fixed. Thanks!