Archive for the ‘19th Century Mathematicians’ Category

I never studied the logistic equation as a student.  I first encountered this relationship as an instructor in one of the College Algebra textbooks I was reviewing and/or teaching from and was intrigued by the application of this “growth with constraints” model to a natural resource.  In researching applying the logistic model to natural resource consumption, I immediately ran into M. King Hubbard’s work on Peak Oil.

Then, when I was teaching integral calculus last winter, we began a unit on separable differential equations.  I was poking around looking for good application problems that would utilize separable ODEs and ran into the fundamental population differential relationship \frac{dP}{dt}=kP, followed by the relationship I had used for the logistic \frac{dP}{dt}=kP(1-\frac{P}{N}) with N defined as the “carrying capacity” or maximum growth for the population.

We went through the procedures for solving each of these ODEs (as well as the continuous mixing problems which follow a similar pattern) and then we moved on.

This year when I was teaching this topic again I was reminded of a paper my thesis advisor had given to me back in 2003 about the application of differential equations to modeling fish populations.  I was intrigued by the profusion of models that could be generated by changing the constraints for a given relationship.

Since I didn’t teach integral calculus for another 10 years after I read that paper, I had essentially forgotten most of the equations, formulas and relationships that generated the graphs that had stuck with me.  This year, while covering the separable ODEs with their applications, I began to look into the application of these relationships to fish populations and population in general.

I found two great resources that go through the set up of these relationships in a very clear manner, and each of them includes wonderful graphs showing the multiple solutions that result when the same differential relationship is solved with different initial conditions.

The opening section of Robert Borelli and Courtney Coleman’s Differential Equations: A Modeling Approach can be read here.

A student project from James Madison University written by Bailey Steinworth, Yuhui Wang and Xing Zhang can be read here.

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I gave a one-hour talk on the The Mathematics of Peak Oil on May 18th.

The power point slides from the talk are here.

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A presentation on M. King Hubbert‘s adaptation of Pierre Verhulst‘s logistic function to model oil production.

The proof that a paraboloid of revolution reflects parallel waves to a single point.

How to calculate distance between two locations on a sphere given latitude and longitude.

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I read a review of the graphic novel Logicomix in the New York Times recently.

Logicomix is a graphic novel that dramatizes the work and lives of some of the most important mathematical logicians of the late 19th and early 20th century.

A review in the Financial Times explains:

First among them is Bertrand Russell, the English philosopher whose life story this is – at least as far as 1939. Also present are his fellow pioneers in the philosophy of mathematics: Alfred North Whitehead, with whom Russell sought, in the years before the first world war, to provide a logically rigorous, good-for-all-time foundation for mathematics; Ludwig Wittgenstein, the austere Austrian who argued that Russell’s project was misconceived; Kurt Gödel, Wittgenstein’s compatriot, who proved that it was; and assorted other pin-ups of higher mathematics – Cantor, Poincaré, Hilbert.

So, essentially, what was going on is that the British logician/mathematician Bertrand Russell, together with his collaborator Alfred North Whitehead, were trying to prove that mathematics was “self-contained.”  That is to say that they were trying to show that there is no “fudge factor” in mathematics.

It is understandable that someone might be sucked into believing that this is possible.  Math can be very reassuring in the apparently solid nature of its logical structure.  It sometimes appears as if there are no holes in the chains of reasoning upon which all of mathematics is built.

But, in reality, mathematics is and always has been a construct that has significant holes in its foundations.  Not being able to divide by zero may be the least of math’s logical dilemmas.

This is not to say that the lack of logical perfection makes mathematics useless.  Far from it, math has been and remains a very useful reasoning tool for humans.  In fact, one of the most amazing things to me about math is that Kurt Gödel used mathematical logic itself to demonstrate the holes that exist in the logical structure of mathematics.

Gödel, Escher and Bach

I was first introduced to these events in the history of mathematical logic through the book Gödel, Escher and Bach, which was written in 1979 by Douglas Hofstadter.

In the introductory chapter, Hofstadter explains the basic ideas of the work of Bertrand Russell and the Austrian mathematician/logician Kurt Gödel.  Russell and Alfred North Whitehead were working very hard in the early 20th century to establish a rock solid logical foundation for mathematics that would dispel all of the paradoxes and conundrums that both ancient and contemporary mathematicians had struggled with.  Russell and Whitehead’s work was eventually published as the three-volume Principia Mathematica.

The simplest example of the type of paradox that Russell and Whitehead were concerned with is the one that Hofstadter gives in Gödel, Escher and Bach:

That paradox is the so-called Epimenides paradox, or liar paradoxEpimenides was a Cretan who made one immortal statement: “All Cretans are liars.”  A sharper version of the statement is simply “I am lying”; or, “This statement is false”.  It is that last version which I will usually mean when I speak of the Epimenides paradox.  It is a statement which rudely violates the usually assumed dichotomy of statements into true and false, because if you tentatively think that it is true, then it immediately backfires on you and makes you think it is false.  But once you’ve decided it is false, a similar backfiring returns you to the idea that it must be true.  Try it!

This paradox is particularly vexing for mathematicians because mathematics relies extensively on the “Law of the Excluded Middle,” which assumes that a statement is either true or false.

In Gödel, Escher and Bach, Hofstadter explains that many mathematicians weren’t sure that Russell and Whitehead would be able to accomplish their task, so the mathematician David Hilbert set up a challenge to mathematicians to determine conclusively whether Russell and Whitehead were right or wrong.

In 1931, the Austrian mathematician Kurt Gödel demonstrated that Russell and Whitehead’s work was incorrect – or, if it was correct, it was incomplete.  This is the essence of Gödel’s Incompletness Theorem.  It says that no formal mathematical system can be both complete (“all-encompassing”) and logically consistent.

I was always curious as to why Russell felt that mathematics could be fully explained through logic, when it seems pretty clear that the world we live in is far more complex than the rules of logic seem to account for.

Logicomix explores Russell’s private life, both as a child and as an adult.  Russell’s family situation as a child was somewhat unsettled and he came from a family that had a history of madness.  It may be that Russell’s love of mathematics and craving for logical certainty was an understandable attempt to ground himself in a family situation that was somewhat unstable.

The scientific and mathematical results of this period of history are important milestones in our culture.  It was in the 1920’s that Werner Heisenberg, Niels Bohr and Albert Einstein were working through the implications of quantum physics and it was in 1935 that Schrödinger first conceived of his cat that was both alive and dead.

These ideas and conclusions put 20th century European and American science and philosophy on shaky ground, and mathematicians and physicists continue today to work through the implications of these ideas.

It is unclear to me whether or not it is possible to resolve the various paradoxes and seemingly nonsensical conclusions that arise out of this body of work.

I think that examining the scientists and mathematicians themselves as historical figures (as Logicomix does) can be helpful in creating a structure for inquiry into an area of knowledge that, almost by definition, lacks structure.

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Analysis does not owe its really significant successes of the last century to any mysterious use of √(-1), but to the quite natural circumstance that one has infinitely more freedom of mathematical movement if he lets quantities vary in a plane instead of only on a line.

Leopold Kronecker, (1894) quoted in Remmert’s Theory of Complex Functions

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Georg Cantor was an important mathematician who lived during the late 19th century.  He devised a method for comparing different sizes of infinite quantities and demonstrated that there were differnet sizes of infinite sets.

The foundation of Cantor’s ideas is the concept of a one-to-one correspondence.  This is a concept which is deeply rooted in the human psyche, as is shown by the fact that the word digit – meaning number – also means finger.  This one-to-one correspondence between enumerating objects by comparing them to the fingers and/or toes of the human body is the presumed origin of our base ten number system.

What Cantor did was to compare infinite sets by establishing a one-to-one correspondence bewteen the elements of the two sets.  If it was possible to establish a one-to-one correspondence, then Cantor concluded that the sets were the same size.  In 1873, Cantor published a paper showing that the rational numbers (which includes all fractions and whole numbers) is actually the same size as the counting numbers (1, 2, 3, …and so on).  Any set that is the same size as the counting numbers is called “countable.”

Another example of this is the set of even numbers.  The set of even numbers is the same size as the set of counting numbers.  It might seem as though there would be fewer even numbers, because the set of counting numbers includes all the odds, but the set of even numbers doesn’t!

But, you can create a one-to-one correspondence between the coutning numbers and the even numbers.

1 (in the counting numbers) matches up with 2 (in the even numbers)

2 (in the counting numbers) matches up with 4 (in the even numbers)

3 matches up with 6

4 matches up with 8 and so on.  Any counting number N will match up with 2N in the set of even numbers.

For every counting number, there is a corresponding even number – therefore the sets are the same size.  Cantor called this size – aleph null, the smallest infinite size.  Cantor showed that the Real Numbers, represented by the number line and including all rational and irrational numbers, are NOT countable – in fact, there are more Real Numbers than Counting Numbers.  Cantor’s famous Diagonal Proof is actually fairly simplistic.

Most mathematicians of the late 19th century hated Cantor’s ideas, and he didn’t receive much support intellectually while he was struggling with his work.  Cantor often became depressed toward the end of his life and spent time in and out of sanitariums, eventually dying in 1918.  By this time, however, many younger mathematicians looked up to Cantor and relied heavily on his work and ideas.

Today Cantor is regarded as one of the most important mathematicians to ever work.  His ideas regarding infinity are deeply philosophical and have had a profound impact on mathematics, science and philosophy.

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