J. Ogden Armour was the son of the Civil War industrialist Philip Danforth Armour and legendary meat packaging robber-baron of his time.

Armour the younger took the company his father had left him and turned it into a global powerhouse. Sales soared from $200 million to over $1 billion *between 1901 and 1919.* Such was his efforts that to this day, when you walk into a super market, it is quite likely you will find the deli meat still packaged under his family name.

In his day, his wealth marked him the second richest man on the planet with a fortune of approximately $100 million. This was equivalent to about $1.5 billion as of today.

His lifestyle was lavish. In addition to a well known 1,200 acre estate at Lake Forest, IL, it is less known that Armour also had an extravagant property in Traverse City, MI, at which this author has had the pleasure of staying. It spanned a vast wilderness, marked by an impressive mansion, a six bedroom house for the groundskeeper, large barn, boat house and lake house, with two private lakes contained on the property.

And then, quite abruptly, it all came to a halt.

Following the end of World War I, Armour made a bold call that German demand for meat would be extensive. Intent on tapping into the market Armour began to direct ships of meat products to German ports.

As report after report came back that the meat was not being purchased, Armour, ever confident that German demand was going to rebound any day, directed yet more ships to carry yet more meat.

Where it sat and rot…

Armour had underestimated the extent to which the war had exhausted Germany and left its people impoverished. Now clear with the benefits of hindsight, the Allied forces had run a heavy toll on Germany and the ensuing war reparations that followed, at the behest of France, left the country in ruin. These same effects would in many ways sow the seeds for the second World War to follow.

Armour’s efforts began to ruin his company, soon $144 million in debt. More convinced now than ever of the coming of German buyers for his meat, Armour began to route his own personal fortune directly into shipping the packaged meat to German shores.

When the smoke cleared the company had lost $125 million in less than three years. In some respects the losses for J. Ogden Armour were far worse. *Armour had personally lost $1 million a day for 130 days.* Bankrupted, the government, which was now outraged to watch one of America’s largest companies undergo such absurd losses from only a bet by its president, began to investigate Armour.

Armour was ousted as president of his namesake company in 1923. The Armour family was permanently broken. Yet Armour’s employees were so loyal to him that none would testify against him. The government eventually dropped its case. J. Ogden Armour died in 1927 with less than $25,000.

Following his death and with the rebound of the stock market, some stock he had retained which had been previously worthless regained a value of some $3 million, which returned his widow to wealthy status. However, the Armour family name was never again to regain the power and prestige it had known in the those early years of the Twentieth Century.

]]>There are just two seminal problems with this pardon. The first is that Alan Turing has been dead 60 years. The second is that he never should have needed “pardon” in the first place.

Each of you is intimately familiar with the work of Alan Turing, although you may not realize it yet.

In the turn of the 20th century, the mathematician David Hilbert proposed mathematics redirect itself to a task worthy of the Zeitgeist of those early years, aiming to spurn mathematical intuition at the alter of the positivists.

The primary aim of the program was to demonstrate that all of mathematics could effectively be replaced by a computer approach to proof and a set of well chosen axioms (without getting into the mostly now-worthless terminology of long forgotten philosophical movements, let’s just call them the “MASTER AXIOMS”). These were the practical aims – however, beneath the surface were other goals. The destruction of metaphysics, the removal of the “mathematics problem” to the positivist worldview, the death of Platonic reasoning, and the emergence of a permanent state of pure-empiricism.

The real tangible quest of the movement was to demonstrate three things: that mathematics was (1) consistent, (2) complete, and (3) decidable.

The first (and in some respects final) blow against the Hilbert program was struck in 1931 with the publication of *Uber formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme* (On formally undecidable propositions of Principia Mathematica and related systems) by Kurt Godel. In his paper, the young Godel brought havoc on the positivists and the Hilbert program, by demonstrating that axiom derived mathematics was both not-consistent and not complete.

However, after the Hilbert program came crashing down, there were remnants and the third element of the program remained unanswered.

It was Alan Turing who helped take down the final pillar of the Hilbert program. He did this by creating the hypothetical machine, known as the “Turing Machine”. A device that could return a true or false answer for any decidable statement, the Turing machine enabled Alan Turing in 1936 to demonstrate critically that all of mathematics is also not decidable. This laid to rest the rampant idealism that had seized mathematics while also helping to demonstrate, alongside the great work of the likes of Godel, Church, or Von Neumann, some of the barriers to which mathematics can be successfully employed.

And it is his machine through which each of you is intimately familiar with Alan Turing as you read this on your electronic devices. The Turing machine was the architectural great grandfather of what would become the modern day computer.

A code breaker in the second World War, Alan Turing was the unsung champion of the triumph of the Allies. Employing his earliest computational devices, he engaged in the immensely complex task of breaking the German enigma – the ingenious pre-computer method of sending coded messages. The information the Allies gained from these decoded messages was of such importance, without it most of the Alan Turing’s home country, the UK, would have been overrun by invading forces otherwise. But by Turing’s hand, the Allies frequently knew of German plans before even German infantry.

At the complete of the war, peace returned in a form to Europe. But not for Turing. No ceremonies or honors were waiting for him. No peace, either.

A homosexual, and the status quo that had carried his necessity now behind them, Alan Turing was prosecuted and convicted in 1952 for indecency and subjected to castration.

He killed himself in 1954.

The loss of Alan Turing was a tragedy. His final work was moving into the subject of complexity theory. Between his sentencing and the time of his suicide, Turing began answering questions in mathematical biology (some of the most complicated around). This path eventually leads towards the direction of genetic algorithms or non-linear systems. Consider that recent papers written between 2012-2013 have concluded that photosynthesis follows many of the same behaviors of quantum mechanics. Turing clearly had the prowess to answer this level of inquiry. How much further along would we be today if he had lived?

I do not know who has pushed to have Turing pardoned by the state. Then prime minister Gordon Brown did formally apologize for the plight of Turing in 2009. However, pardon is nonsense. Alan Turing needs no pardon – you pardon criminals as an act of forgiveness. But the sin rests squarely on the shoulders of the UK. If they really want to make amends they should denounce the original conviction, and drag the names of those that persecuted this titan of the mind through the mud, striking them of any honors.

Then on the nation’s behalf, they can beg the memory of Alan Turing for pardon.

]]>Georg Ferdinand Ludwig Philipp Cantor was a German mathematician who lived from around the middle of the 19th century to the turn.

Of Georg Cantor’s mathematical discoveries, he is most famous for developing set theory – an approach to solving proofs related to the span of all mathematical studies and wildly popular in the mathematics circles. His notation was so popular, that for a brief spell, pre-Gödel, some believed Cantor had discovered a means of proving or disproving any statement or proposition one could make.

But while Cantor’s set theory was popular, one of his greatest discoveries was so poorly received that the details of its reception are a matter of notoriety amongst mathematicians.

The subject was on what today is called transfinite numbers; and Cantor’s conclusion was something of the following:

Suppose I take all the positive real numbers 1,2,3…n and lay them in a set unto themselves and denote this R. And suppose that next to that set I take all the numbers between 1 and 2 and put them into a set and denote that S.

Which of these two sets of numbers is bigger?

After much deliberation, Cantor finally came to the conclusion that both had to be the same size. Thus, he developed the concept of ~~finitely~~ countably infinite sets and ~~infinitely~~ uncountably infinite sets. Between any two numbers there exist an infinite number of other numbers.

A second of Cantor’s controversies was around the following problem. Which is bigger, the set of all real numbers from (…-3,-2,-1,0,1,2,3…), or the set of only the positive real numbers (1,2,3…)?

The widely accepted answer was that the first collection of numbers must be bigger than the second, because the second set is contained within the first.

Cantor proved otherwise.

Cantor argued that for any of these sets, there exists an endless number of other numbers to make up the set. As both sets extends to infinity, it is formally impossible to reason that the first set is bigger than the second.

The first set contains the second set. **Yet they are the same size.**

This caused an uproar. Poincare and Kronecker (two extremely accomplished mathematicians in their own right) launched an assault on Cantor that crossed from merely objective to the highly personal. Other mathematicians eventually entered the attack also.

Then the Christians got involved. They seemed to feel that Cantor’s work was a direct assault on the omnipotence of God, a charge Cantor would spend his entire life fighting.

Cantor suffered a nervous breakdown in 1884 and was hospitalized. Upon release, he spent the next few years trying to piece together his reputation and make amends with his detractors. In 1899, he was checked into a sanatorium. Then again in 1903. In 1904, a paper from the mathematician Konig attempting to disprove his theorem, read out loud in front of his family, left him so thoroughly demoralized, he is said to have questioned the existence of God over his treatment. This set off a string of stays in mental hospitals that spanned every two to three years.

He retired from mathematics in 1913, in a state of abysmal poverty. He suffered a final mental breakdown and died in 1918. The effort of contemplating the infinite cost him his career, his reputation, and his very mind.

Shortly thereafter, his detractors finally admitted they could find no means of disproving his central theorem of transfinite sets. Today, it is so readily accepted that it is taught in undergraduate courses and occasionally introduced as early as grade school.

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