Monthly Archive: January 2015

I suppose that this post grew out of a recent opportunity I had to re-watch the M*A*S*H television series. One particular episode, entitled The Light That Failed, finds the army surgeons and nurses of the 4077^th unit suffering a brutal Korean winter and desperately low on supplies.  The supply truck soon arrives bearing a cargo more suited for a military unit in the Guam or the Philippines and not mainland Asia.  As the troops are left wondering what they can do with mosquito netting, ice cream churns, and swim fins when the temperature is hovering around freezing, someone notices that one of the doctors, B J Hunnicutt, has received a very rare object – a book.

And this is not just any book, but a murder mystery by the famed writer Abigail Potterfield called the Rooster Crowed at Midnight.  Either out of the goodness of his heart or out of a desire to end the constant nagging (probably both), B J decides to tear portions of the book out so that it can circulate throughout the camp.  As the old saying goes, no good deed goes unpunished, and soon he discovers that the last page of the book, in which all is revealed, is missing.  And thus begins the great debate as to who committed the murders, how they did it, and why.

The team comes up with many answers, all of which are first widely embraced as the solution and then scuttled when someone gives a counterexample that pokes a hole in the theory.  They eventually place a long distance phone call to the author herself, now residing in Australia, to get the answer.  But even this authoritative voice doesn’t quell the skepticism.  Shortly after they ring off, Col. Potter, the commanding officer, points out that Abigail Porterfield’s own solution can’t be true.  The episode closes with the delivery of the much-needed supplies, and some comic hijinks, but with no satisfactory explanation as to who the culprit was.

I was in middle school when I first saw that episode and it left a lasting impression on me.  For many years I carried misconceptions about mystery stories and I wondered why anyone would ever read them.  In particular, I held a very skewed idea about deductive reasoning and what can and cannot be determined from the evidence.  With the perspective of years (really decades) I am both happy and disappointed to say that I was not alone in my poor understanding of what logic and reason are capable of achieving.

Let’s talk about deductive logic first.  The basic idea behind deductive logic is that the conclusion is infallible if the premises are true.  It is a strong approach to logic, since it argues from first principles that apply to a broad class or set of objects and, from those, narrows down a conclusion about a specific object.  In a pictorial sense, deductive logic can be thought of in terms of Venn diagrams.  If we want to conclude something about an object we simply need to know into what categories or classes this object falls and we will be able to exactly conclude something about it by noting where all of the various categories to which it belongs intersect.

Deductive reasoning is, unfortunately, also limited by the fact that we are not born with nor does anyone have the universe’s user manual that spells out in detail what attributes each object has and into what categories they may be grouped.  So, the standard objections that are raised in deductive logic fall squarely on disagreements about the truth of one or more premises.

For example, the syllogism

is a logically correct deduction, since the conclusion follows from the premises and it is true since the premises are true (or at least we regard them to be true). The syllogism

is invalid, even though its premises are true, since it argues from the specific to the general.  In contrast, the syllogism

is perfectly valid, since the conclusion follows from the premises, but is not true since neither premise is true (or so I hope!).

All of this should be familiar.  But what to make of this ‘deduction’ (B J’s syllogism) made by B J Hunnicutt in the episode:

Is this really a deduction as he claimed?  According to the novel, the first three premises are true.  The fourth premise is certainly plausible but is not necessarily true.  How then should we feel about the conclusion?  What kind of logic is this if it is not deductive?  Suppose we knew that Randolf was the murderer (e.g., we caught him in the act). What can we infer about the fourth premise?

Before answering these questions, consider what would happen if we were to modify the argument a bit to simplify the various possibilities.  The syllogism (B J’s syllogism revised) now reads

This argument is certainly a stronger one than the first one proffered, but it isn’t really conclusive.  But, again, how should we feel about the conclusion?  What kind of logic is this?

In both cases, we know that the conclusion is not iron-clad; that it doesn’t necessarily follow from the premises.  But just like those fictional characters in M*A*S*H, we are often faced with the need to draw a conclusion from a set of premises that do not completely ‘nail down’ an unequivocal conclusion.

The type of logic that deals with uncertainty falls under the broad descriptions of inductive and abductive reasoning.  Inductive reasoning allows us to draw a plausible conclusion ‘B’ from a set of premises ‘A’ without ‘B’ necessarily following from ‘A’.  Abductive reasoning allows us to infer the premise ‘A’ based on our knowledge that outcome ‘B’ has occurred.

In the M*A*S*H examples given above, B J’s revised syllogism is an example of inductive reasoning.  All the necessary ingredients are there for Randolf to have committed the crime but there is not enough evidence to inescapably conclude that he did. We can infer that Randolf is the killer but we can’t conclude that with certainty.

B J’s original syllogism is a lot more complicated.  It involved elements of both inductive and abductive reasoning.  If we believe Randolf is guilty, we might then try to establish that there were secret passages in Huntley Manor that connected the locked library to some other room in the mansion.  We would then have to also establish, maybe through eyewitness testimony, that Randolf knew of the passages (e.g., an old servant recalls showing it to a young Randolf).  Even still, all we would be doing is establishing the premises with more certainty.  The conclusion of his guilt would still not necessarily follow.  If, on the other hand, we knew that he was guilty, perhaps he was seen by someone looking into the library from outside, we might abductively infer that there was a secret passage and that Randolf knew of its existence.

So here it is, a great irony of life.  It’s decades after I first watched an episode of M*A*S*H that turned me off of mystery stories for a long time, and I find myself using that episode as a model for discussing logic and reason.  That, I can’t figure out.

“How language shapes thought and thought shapes language” is an age old question in linguistics and philosophy.  I’m not in any position to give a definitive answer, nor, I suspect, is anyone else.  Having taught math and physics at the university level, I am willing to offer some thoughts about how the language of mathematics and the symbols and glyphs used to turn mathematical concepts into written words shape how people think and solve problems.

In this blog I will be focusing in the concept of infinity and the philosophic implications that come from using it.  But before I get to infinity directly, I would like to discuss, by way of a warm-up exercise, how the use of the symbol ‘x’ throws off a lot of beginning students.

When describing a function or mapping between two sets of real numbers, without a doubt, the most common notation that teachers use is to allow the symbol ‘x’, called the independent variable, to be any member of the initial set, and the symbols ‘y’ and ‘f(x)’ to be the corresponding member of the target set and the function that generates it.  The symbolic equation ‘y = f(x)’ becomes so rigidly fixed in some students minds, that the idea that the symbols ‘x’ or ‘y’ could be replaced with any other symbol, say ‘y’ and ‘z’, never occurs to them.  I myself have experiences of students coming and asking if their book has a typo when it asks them to solve ‘x = f(y)’ or integrate ‘f(y) dy’ or the like (once this happened while I was out to dinner at Olive Garden with my family – but that is a story for another day).

There is no easy way to fix this problem as there is a kind of catch 22 in the teaching of mathematics.  One on hand, the mapping between sets exists as a pictorial relation between ‘clouds’ and ‘the points within them’

without the need for written glyphs.  One the other hand, an initial set of well-defined symbols keeps initial confusion to a minimum and allows the student to focus on the concepts without all the possible freedom of choice in notation getting in the way.  (Note:  a reader comfortable with classic philosophy may point out that a mapping between sets can be abstracted even further, perhaps to the notion of a Platonic form, but this is a side issue.)

Okay, with the appetizer firmly digesting in our minds, let’s turn to perhaps the most confusing symbol in all of mathematics, the symbol for infinity, ‘$\infty$’. This symbol, which looks like the number ‘8’ passed out after a night of heavy drinking, seduces students and instructors alike into all sorts of bad thoughts.

How does it have this power, you may ask? Well, its very shape, small and compact and slightly numberish, encourages our minds to treat it like all other numbers. There are literally countless examples of infinity masquerading as a plain number, much like a wolf in sheep’s clothing. One of the most egregious examples is the innocent-looking expression

\[ \int_0^\infty e^{-x^2} dx = \frac{\sqrt{\pi}}{2} \]

where ‘∞’ is compared side-by-side with the number ‘0’. There is perhaps a more palatable way of writing the integral as

\[ \lim_{a \rightarrow \infty} \int_0^a e^{-x^2} dx = \frac{\sqrt{\pi}}{2} \]

but it still looks like the number ‘a’ can be thought of as becoming or approaching ‘∞’. A seasoned practitioner actually knows that both expressions are really shorthand for something much more involved. I will summarize what this more involved thing is in one short and sweet sentence. Infinity is a process that you have the freedom to perform as many times as you like. Or even shorter: Infinity is an inexhaustible process.

Take a moment to think that through. Are you back with me? If you don’t see the wisdom in that maxim consider either form of the integral expression listed above. In both cases, what is really being said is the following. Pick an upper bound on the integral (call it ‘a’ to be consistent with the second form). Evaluate the integral for that value of ‘a’. Record the result. Now increase ‘a’ a bit more, maybe by doubling it or multiplying it by 10 or however you like, as long as it is bigger. Now evaluate the integral again and record the result. Keep at it until one of several things has happened: 1) the difference in the recorded values has gotten smaller than some threshold, 2) you run out of time, or 3) you run out of patience and decide to go do something else. The term infinity is simply meant to say that you have the freedom to decide when you stop and you also have the freedom to resume whenever you like and continue onwards.

If you are new to calculus, you will no doubt find this short sentence definition somewhat at odds with what your instructors have told you. Where are all the formal limits and precise nomenclature? Where is all the fancy machinery? If you are an old hand at calculus you may even be offended by the use of the words ‘process’, ‘freedom’, or ‘inexhaustible’. But this sentiment is exactly at the heart of the Cauchy delta-epsilon formalism, and the casual nomenclature has the advantage of ruthlessly demolishing the ‘high-brow’ language of mathematics to bring what is really a simple idea back to its rightful place as an everyday tool in the thinking person’s toolbox.

On the other hand you may be thinking that everyone knows this and that I am making a mountain out of a mole hill. If you fall into that camp, consider this video about the properties of zero by the Numberphiles.

I must admit I like many of the Numberphile’s videos, but this one made me shake my head. They allowed language to affect their thinking, and they were seduced by the evil camouflage powers of infinity. They go to great trouble to explain why you can’t divide by zero, and they note that people say “isn’t dividing by zero just infinity?” and they point out it isn’t that simple.

The problem is, it is that simple! Dividing by zero is infinity as understood by the maxim above. The Numberphiles prove this fact themselves. At about a minute into the video, one of their lot begins to explain how multiplication is ‘glorified addition’ and division is ‘glorified subtraction’. The argument for ‘glorified subtraction’ goes something like this.

If one wishes to divide 20 by 4, then one keeps subtracting 4 until one is lefts with a number smaller than 4 (in this case zero). The number of times one engages in this subtraction process is the answer, with whatever piece left over being the remainder. So dividing the number 17 by 5 is a shorthand for subtracting 5 from 17 three times and finding that one has 2 leftover. So one then says 17/5 = 3 with a remainder of 2.

The same bloke (I use bloke because of their cool English or Australian or whatever accents) then says that 20 divided by 0 goes on forever because each time you subtract 0 you are left with 20. Here then is the inexhaustible process that lives at the very heart of infinity. Sadly, while he looks like he is about to hit the bull’s-eye (at 2:20 he even says infinity isn’t a number), his aim goes horribly awry at the last moment when he objects to saying that the expression ‘1/0 = ∞’ can’t be true because one could then go on to say ‘1/0 = ∞ = 2/0’ from which one can say ‘1=2’.

This is, of course, a nonsensical objection since the expression ‘1/0 = ∞’ is a shorthand for saying ‘the glorified subtraction of 0 from 1 (in the sense used above) is an inexhaustible process.’ It is no more meaningful to say that this process is the same as the ‘glorified subtraction of 0 from 2’ as it is to say that ‘1/0’ is the same as any other inexhaustible process, like halving a non-zero number until you reach zero.

The fact that the words ‘0’, ‘1’, and ‘∞’ and the sentence ‘1/0 = ∞’ result in an illogical conclusion is an important warning about the power of language to shape thought. The Numberphile guys had all the right ideas but they came up with a wrong result.

In the last post we discussed the epistimological divisions in philosophy between a priori and a posteriori knowledge and the divisions due to Kant between the notions of analytic and synthetic statements.  As a brief reminder, a priori knowledge stems from first principles and can be understood using the human capacity to grasp the essential nature of things.  A posteriori knowledge is obtained only after examining a thing and coming to a conclusion about its nature – a conclusion that cannot be grasped by reason alone.  An analytic statement is one which is true and in which the subject contains the predicate (that is to loosely say that one defines the other) while a synthetic statement is one that is neither false nor is analytic.

On the surface there seems to be such a strong tie between a priori knowledge and analytic statements, on one hand, and between a posteriori knowledge and synthetic statements, on the other, that there is a temptation to equate the two concepts in each case.  Thus one might want to say that all statements of a priori knowledge are analytic and all statements of a posteriori knowledge are synthetic.

But as is usually the case with logic when examined very carefully, ideas that seem rock-solid based on a casual examination become a lot more uncertain when looked at more thoroughly.  However, these kinds of abstract examinations are often dry.  So for this post we’ll try to apply these ideas to the popular medium of the murder mystery.

What should be said about the murder mystery?  I think that if Aristotle were alive today one of his favorite past times would be reading and/or writing murder mysteries.  This should come as no surprise since Aristotle is credited with formalizing logic and logic and solving mysteries go hand-in-hand.  The murder mystery, or detective story as it also called (not all the crimes are murders – only the most enjoyable ones), are individual studies in epistemology.  At its heart is the idea of pronouncing a statement of truth; of disclosing ‘whodunnit’.

Consider the analysis of G. K. Chesterton, one of the twentieth century’s most profound thinkers and prolific authors, who penned dozens of works on analysis, philosophy, and social criticism.   Chesterton, who was home with logic and critical thinking in its many forms, was particularly fond of the detective story and wrote often about it.  One of his notable observations was:

Rex Stout, the author of over 70 detective stories, had the following very nice description of the detecting process.  Speaking about his gourmand and rotund detective Nero Wolfe, Archie Goodwin, Wolfe’s assistant, has this to say about his boss’s moments of genius:

If a detective story is an individual study in epistemology then it should be possible to examine each detective in terms of their where they fall in the division between a priori and a posteriori knowledge and between analytic and synthetic statements of truth.  In this way, maybe we can shed some light on the thornier sides of this debate and also have some fun doing it.

Before examining some of the great literary detectives, let me state that none of them are purely one way or another.  There is no author of detective fiction (at least not one I would want to read) who would believe that crime can be solved purely by thinking about the world from first principles nor who would believe that crime can be solved solely by the dry gathering of facts.  It is the interplay between the two extremes that is the engine of discovery and truth detection.  Nonetheless, each these detectives leans, as does the author who sits behind their adventures, more towards one extreme or another.

We can envision a categorization scheme for detectives where each is placed on a two-dimensional grid.  To the left is the extreme of the synthetic and to the right the extreme of the analytic.  At the bottom is a posteriori knowledge whereas at the top is a priori knowledge.  An empty grid looks like

and placing a detective in the top right means that he depends more heavily on analytic a priori methods to solve crime than by other means.

Our task is then to debate, and argue, and wrestle with where to place each.  I won’t pretend to have a well-conceived and impregnable argument for what I present below.  Rather I offer it as food for thought and, perhaps, the basis of some really enjoyable discussions with family and friends.

The easiest place of start is with Sherlock Holmes.  For this discussion, I will be dealing only with Holmes in his original incarnation as conceived of by Since Sir Arthur Conan-Doyle and not some of the more modern adaptations.  The sleuth of 221B Baker Street often solved the mysteries confronting him through observations correlated with dry or obscure facts.  Red clay from a particular quarry in northern England combined with an encyclopedic knowledge of the British Rail time tables were a more common route to the solution than ponderings about human nature.  Thus we can classify him as predominantly as synthetic and a posteriori.

The two famous creations of Agatha Christie, Hercule Poirot and Miss Marple, are cut from a decidedly different cloth.  Both of these sleuths depended heavily on their knowledge of human nature and often worked from motive to solution.  Clearly they are both analytic, but it seems to me that Poirot starts more from well-articulated first principles and methodical deduction than his female counterpart.  Poirot can explain exactly how he arrived at his conclusions (consider his ‘mentoring’ of Doctor Sheppard in the Murder of Roger Ackroyd) even if he often won’t and he needs only the bare facts to proceed (The Disappearance of Mr. Davenheim).  In contrast, Miss Marple relies on a lifetime spent examining human nature ‘under a microscope’ in her village of St. Mary Mead.  As she explains in Sir Henry Clithering, her knowledge is akin to an Egyptologist who, due to a lifetime handling Egyptian scarabs, can tell when one is genuine while another is a cheap knockoff even if he can’t explain how.  She often jumps to the solution and then gathers or reconciles facts only latter (Death by Drowning).  Thus I would be inclined to place Poirot in the analytic and a priori sector and Miss Marple in just below him somewhat in the a posteriori square.

Nero Wolfe, already mentioned above, is more difficult to place.  He seems to slide back and forth between the extremes, having the greater fluidity early on in Stout’s writing.  In some cases, he is clearly synthetic in his approach.  Consider Fer De Lance, where he asks a golf club salesman to demo how to swing a club to confirm his suspicions about the delivery method of a poison dart or The Rubber Band, where he realizes a connection between two usages of the word ‘rubber’ to impeach the murderer’s alibi.  In other cases, including The Christmas Party and Death of a Doxy, he relies solely on his understanding of human nature and his ability to play upon a murderer’s irresistible compulsion to force a conviction.  I place him nearly equally balanced between analytic and synthetic and tipping more towards a posteriori than a priori.

The final two detectives I’ll discuss both happen to be Roman Catholic priests: Father Brown the creation of G. K. Chesteron and Brother William of Baskerville from Umberto Eco’s brilliant novel The Name of the Rose.  There is some irony here in that Chesterton was a devout catholic and Eco is a self-declared atheist.  Nonetheless, both detectives depend on their training in philosophy (with particular emphasis on Thomas Aquinas) and the intellectual and theological traditions of the Catholic Church to find solutions to their mysteries.  Father Brown is deeply logical and staunch defender of reason (The Blue  Cross) but is prone to inspired deductions where, as Chesteron puts it (The Queer Feet):

In contrast, Brother William seems to take a more measured approach.  On one hand he is quite proud and comfortable in his use of logic as in the affair of Brunellus the horse as he and Adso, his novice, approached the unnamed abbey where the bulk of the book is set.  At other times, he seems to despair of ever knowing anything or, at least, anything with certainty as in his explanation to Adso of how he got the right answer using from the wrong approach.  (An aside: the whole discussion associated with penetrating and navigating the labyrinth is delightful reading and worth studying).

All things considered, I tend to plop Father Brown down into that controversial region where synthetic a priori knowledge sits and I place Brother William firmly in the center.

My final diagram looks like:

Obviously, I’ve ignored a host of beloved literary detectives, including C. Auguste Dupin, Perry Mason, Ellery Queen, Lord Peter Wimesy, and Sam Spade.  Leave a comment telling where on the diagram you placed your favorites and why.

I suppose that the genesis of this post comes from one of my current study projects.  Over the past several months, I’ve been slowly working my way through Harry Gensler’s really fine book ‘An Introduction to Logic’, 2^nd edition.  As is the case when I learn anything, I find that my mind automatically associates many things with many things.  It seems to me a good strategy, because I remember the information much better and can apply it with greater ease.  (This should be contrasted with the way I was taught or learned history – I still don’t know what the Battle of Hastings was, why I should care, and how it affects my life.)

Anyway, Chapter 3 of Gensler’s book deals with definitions and what is essentially epistemology, although I don’t believe that Gensler ever mentions that term explicitly. The most interesting part of that discussion is the presentation of the categories of definition attributed to Immanuel Kant and how they mesh with the two philosophical divisions of knowledge that are traditionally recognized.

Kant divides definitions into two categories:

Traditionally, philosophers recognize two kinds of knowledge, which are defined as:

No doubt a few examples are in order to make these concepts clearer.  The examples that Gensler provides (and which I believe an anonymous Wikipedia contributor lifted without attribution) tend to feature the noun ‘bachelor’.

Examples of analytic and synthetic statements are:

The first statement is analytic, since its subject ‘bachelors’ is synonymous with ‘unmarried’ (that is to say that its subject contains its predicate as an attribute), while the second statement is clearly synthetic, since the word ‘Daniel’ is not synonymous with ‘bachelor’, nor is it self-contradictory, as it would be if ‘Daniel’ were replaced by ‘Stacey’ (assuming the usual gender denotations of names).

The following statements are examples of a posteriori and a priori knowledge:

The first piece of knowledge that ‘some bachelors are happy’ can only be obtained by us going out, meeting bachelors and determining (through whatever mechanism we like) that they are happy.  The second bit of knowledge is based on our ability to see the essential definition of the word bachelor.

Obviously, there is an extremely close tie between a statement being analytic and a piece of knowledge being a priori.  There is also a very close tie between a synthetic statement and a piece of a posteriori knowledge (but, I would argue, not as close as the association between analytic and a priori).  Thus, there is a tendency in philosophy to equate the two terms in each case, and to say that all statements of a priori knowledge are analytic, and that all statements of a posteriori knowledge are synthetic.

This seems to be a natural conclusion, and one may dismiss the idea that some statements of a priori knowledge can be synthetic, or that some statements of a posteriori knowledge can be analytic. This dismissal is also supported, at least superficially, by the common notion that all of our mathematics is a priori knowledge and all of our science is based on a posteriori knowledge.

The problem arises when one starts to examine certain statements that, while not quite self-referential, fall into a category where they at least talk about each other, or, more precisely, they are statements that explicitly talk about the nature of knowledge.

As a possible example of an analytic statement of a posteriori knowledge, consider the sentence ‘the value of pi is about 3% larger than 3’.  That there is a constant of proportionality between the diameter and the circumference of a circle is certainly an analytic statement of a priori knowledge, but the determination of the actual value (or some decimal approximation to it) is not.  Okay, so maybe there is such a thing as an analytic statement of a posteriori knowledge, although Gensler leaves the door open for doubt when he says

But, apparently, the real drama in the philosophical world (I must admit I have fanciful images of Plato and Aristotle, dressed in wrestling tights, as squaring off in a steel-cage match) is over whether there is credible evidence to support the claim of a synthetic statement of a priori knowledge.  Such a statement Q would be one such that Q is neither self-contradictory to affirm nor to deny, Q is true, and we know Q to be true only using our reason.

Trying to further explain where such a brain-twisting idea can arise, Gensler asks us to consider two types of philosophers: empiricists and rationalists.  According to his discussion, the empiricist denies the possibility of synthetic a priori knowledge, while the rationalist admits such a possibility.  The crux seems to come in the examination of the empiricist’s point of view.  The first observation is that an empirical point of view seems to equate the experiences of the senses with the actualities of the world.  An empiricist is inclined to say something like

Of course the empiricist seems to have no mechanism for embracing the idea that an object is actually red when it is perceived as red, except to resort to what seems to be synthetic a priori knowledge.  It is synthetic because nothing in how the terms are defined requires an object that is perceived as red to actually be red.  It is a priori because we use our reason to conclude that it is a tenable assumption that all objects perceived as red are, indeed, red.

Perhaps even more interesting is the position the empiricist takes on synthetic a priori knowledge in the first place.  To say

seems to be an example of synthetic a priori knowledge, at least in-so-far as one is willing to agree that the statement, if true, is not true by virtue of the definition of the terms ‘synthetic’ and ‘a priori’, and is therefore synthetic, and that the statement, if true, cannot be determined to be so by our sense experiences, and so it must be a priori.

Okay, so what does any of this have to do with murder mysteries?  Well, as I mentioned above, whenever I am learning something, I employ a personal strategy of associating things I understand with things I am trying to grasp.  As I was reading Genler’s presentation, I couldn’t help but wonder how mystery writers employ these points to amuse, entertain, and sometimes baffle us.

So, next time, I will apply some of these concepts to some of the world’s most famous fictional detectives.  We’ll have a chance to see if Sherlock Holmes is synthetic or analytic.  We’ll ask how many of Hercule Poirot’s little gray cell depend on a priori versus a posteriori knowledge.  We’ll examine whether Miss Marple’s understanding of human nature springs from analytic a posteriori knowledge.  And we’ll explore how logic, reason, and epistemology figure into two of the twentieth century’s most philosophical writers, G.K. Chesteron and Umberto Eco, through their excellent characters of Father Brown and Brother William of Baskerville.