“Kolmogorov – Poincaré – Gauss – Euler – Newton, are only five lives separating us from the source of our science”. – Vladimir Arnold

These are men whose work gives us the tools for our daily lives. Their mastery of numbers is a necessity for modern technological civilization: we should at least be able to say ‘thank you!’

But of course, they are to be Officially Forgotten: wrong sex, wrong race.

Fortunately, I am simply uncaring of such things: even of the agnosticism/atheism of some. They helped humanity, which is made in the Image of God: their actions which helped humanity should therefore honoured, even if the men themselves have no place in the kingdom of God.

Just as Christians should be punished should they unjustly harm people in the here and now, even if their eternal destiny is sure…. as King David knew from experience.

(As seen in Matthew 25:31-44, actions *matter*.)

Now, regarding the men themselves:

Over two miraculous years, during the time of the Great Plague of 1665-6, the young Newton developed a new theory of light, discovered and quantified gravitation, and pioneered a revolutionary new approach to mathematics: infinitesimal calculus.

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Despite being by far his best known contribution to mathematics, calculus was by no means Newton’s only contribution. He is credited with the generalized binomial theorem, which describes the algebraic expansion of powers of a binomial (an algebraic expression with two terms, such as

a^{2}–b^{2}); he made substantial contributions to the theory of finite differences (mathematical expressions of the formf(x+b) –f(x+a)); he was one of the first to use fractional exponents and coordinate geometry to derive solutions to Diophantine equations (algebraic equations with integer-only variables); he developed the so-called “Newton’s method” for finding successively better approximations to the zeroes or roots of a function; he was the first to use infinite power series with any confidence; etc.

Not bad for a biblical literalist. Now, if only *today’s* biblical literalists would actually push forward our mastery of the universe, as Newton did.

“But it’s so easy to run and hide and dream of the Second Coming… any moment now! Much more easier than doing the actual work needed to extend Christ’s Kingdom, in the here and now!”

Despite a long life and thirteen children, Euler had more than his fair share of tragedies and deaths, and even his blindness later in life did not slow his prodigious output – his collected works comprise nearly 900 books and, in the year 1775, he is said to have produced on average one mathematical paper every week – as he compensated for it with his mental calculation skills and photographic memory (for example, he could repeat the Aeneid of Virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last).

Today, Euler is considered one of the greatest mathematicians of all time. His interests covered almost all aspects of mathematics, from geometry to calculus to trigonometry to algebra to number theory, as well as optics, astronomy, cartography, mechanics, weights and measures and even the theory of music.

Much of the notation used by mathematicians today – including

e,i,f(x), ∑, and the use ofa,bandcas constants andx,yandzas unknowns – was either created, popularized or standardized by Euler. His efforts to standardize these and other symbols (including π and the trigonometric functions) helped to internationalize mathematics and to encourage collaboration on problems.He even managed to combine several of these together in an amazing feat of mathematical alchemy to produce one of the most beautiful of all mathematical equations,

e^{iπ}= -1, sometimes known as Euler’s Identity. This equation combines arithmetic, calculus, trigonometry and complex analysis into what has been called “the most remarkable formula in mathematics”, “uncanny and sublime” and “filled with cosmic beauty”, among other descriptions. Another such discovery, often known simply as Euler’s Formula, ise^{ix}= cosx+isinx. In fact, in a recent poll of mathematicians, three of the top five most beautiful formulae of all time were Euler’s. He seemed to have an instinctive ability to demonstrate the deep relationships between trigonometry, exponentials and complex numbers.

Do I have even the vaguest idea of Euler’s Identity? Well, a vague idea yes, but my mathematical abilities are far too weak to grasp the real significance of it.

But I certainly DO understand the significance of mathematical notation, and I can respect the determination of a man who did not let his talents go to waste, but put them to work, regardless of blindness or tragedy.

Why on earth didn’t I hear of him until today?

At 15, Gauss was the first to find any kind of a pattern in the occurrence of prime numbers, a problem which had exercised the minds of the best mathematicians since ancient times. Although the occurrence of prime numbers appeared to be almost competely random, Gauss approached the problem from a different angle by graphing the incidence of primes as the numbers increased. He noticed a rough pattern or trend: as the numbers increased by 10, the probability of prime numbers occurring reduced by a factor of about 2 (e.g. there is a 1 in 4 chance of getting a prime in the number from 1 to 100, a 1 in 6 chance of a prime in the numbers from 1 to 1,000, a 1 in 8 chance from 1 to 10,000, 1 in 10 from 1 to 100,000, etc). However, he was quite aware that his method merely yielded an approximation and, as he could not definitively prove his findings, and kept them secret until much later in life.

In Gauss’s annus mirabilis of 1796, at just 19 years of age, he constructed a hitherto unknown regular seventeen-sided figure using only a ruler and compass, a major advance in this field since the time of Greek mathematics, formulated his prime number theorem on the distribution of prime numbers among the integers, and proved that every positive integer is representable as a sum of at most three triangular numbers.

Although he made contributions in almost all fields of mathematics, number theory was always Gauss’ favourite area, and he asserted that “mathematics is the queen of the sciences, and the theory of numbers is the queen of mathematics”. An example of how Gauss revolutionized number theory can be seen in his work with complex numbers (combinations of real and imaginary numbers).

Sadly, there is a major moral failing with the man…

As Gauss’ fame spread, though, and he became known throughout Europe as the go-to man for complex mathematical questions, his character deteriorated and he became increasingly arrogant, bitter, dismissive and unpleasant, rather than just shy. There are many stories of the way in which Gauss had dismissed the ideas of young mathematicians or, in some cases, claimed them as his own.

But this is not a list of perfect men (a very short list, with exactly one name), but a list of men who led major advances in our understanding of mathematics. Without these men – even the badly flawed ones – we would be far, far poorer than we are today.

In contrast, what contribution has any of the humanities have made – especially after they were taken over by the Tolerant Ones, who utterly despise the essence of the Western advances: a Single Creator God, with a Single Universal Law, that *can* be largely mastered by men – and *should* be mastered by them!

Paris was a great centre for world mathematics towards the end of the 19th Century, and Henri Poincaré was one of its leading lights in almost all fields – geometry, algebra, analysis – for which he is sometimes called the “Last Universalist”.

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It was one such flash of inspiration that earned Poincaré a generous prize from the King of Sweden in 1887 for his partial solution to the “three-body problem”, a problem that had defeated mathematicians of the stature of Euler, Lagrange and Laplace. Newton had long ago proved that the paths of two planets orbiting around each other would remain stable, but even the addition of just one more orbiting body to this already simplified solar system resulted in the involvement of as many as 18 different variables (such as position, velocity in each direction, etc), making it mathematically too complex to predict or disprove a stable orbit. Poincaré’s solution to the “three-body problem”, using a series of approximations of the orbits, although admittedly only a partial solution, was sophisticated enough to win him the prize.

But he soon realized that he had actually made a mistake, and that his simplifications did not indicate a stable orbit after all. In fact, he realized that even a very small change in his initial conditions would lead to vastly different orbits. This serendipitous discovery, born from a mistake, led indirectly to what we now know as chaos theory, a burgeoning field of mathematics most familiar to the general public from the common example of the flap of a butterfly’s wings leading to a tornado on the other side of the world. It was the first indication that three is the minimum threshold for chaotic behaviour.

Paradoxically, owning up to his mistake only served to enhance Poincaré’s reputation, if anything, and he continued to produce a wide range of work throughout his life, as well as several popular books extolling the importance of mathematics.

The man believed in integrity: something that is in steep decline among today’s scientists. Those who are not politically driven are driven by cash, and the rot is getting deep into the peer review process.

Knowing the laws of the universe demands the recognition that there are laws, that reality exists, that meaning and truth actually exists, that Universal Wrong and Universal Right is active. But the increasing understanding that such things point to a creator has sent Our Modern Intellectual Class into the meaningless (but God-free!) land of the multiverse, where proving and disproving theories is simply unnecessary.

(Just another gift of Darwinism, naturally. “Never mind if it can be disproved or not, nevermind if our universe, grounded in information, demands an intelligent creator and not random chance over billions of years: Evolution the ONLY way to escape God, so it MUST be true!”)

The failure of those who hate God opens the door to those who love Him. And it starts with the commandments, with integrity, with the religious devotion to the truth.

It then goes on to commitment, to the willingness to put in the years, the decades, needed to master the material, and then expand upon it.

If it was up to the Establishment, we would all die in ignorance, so long as the Right Sort, the Certified Experts and Responsible Politicians, stayed in power. Fortunately, it isn’t so hard for believers to get the information they need, and motivated children who respect both truth and beauty – but truth, first and foremost – can continue to push forward our understanding of the universe.

Kolmogorov

Kolmogorov was one of the broadest of this century’s mathematicians. He laid the mathematical foundations of probability theory and the algorithmic theory of randomness and made crucial contributions to the foundations of statistical mechanics, stochastic processes, information theory, fluid mechanics, and nonlinear dynamics. All of these areas, and their interrelationships, underlie complex systems, as they are studied today.

Kolmogorov graduated from Moscow State University in 1925 and then became a professor there in 1931. In 1939 he was elected to the Soviet Academy of Sciences, receiving the Lenin Prize in 1965 and the Order of Lenin on seven separate occasions.

His work on reformulating probability started with a 1933 paper in which he built up probability theory in a rigorous way from fundamental axioms, similar to Euclid’s treatment of geometry. Kolmogorov went on to study the motion of the planets and turbulent fluid flows, later publishing two papers in 1941 on turbulence that even today are of fundamental importance.

In 1954 he developed his work on dynamical systems in relation to planetary motion, thus demonstrating the vital role of probability theory in physics and re-opening the study of apparent randomness in deterministic systems, much along the lines originally conceived by Henri Poincare.

In 1965 he introduced the algorithmic theory of randomness via a measure of complexity, now referred to Kolmogorov Complexity. According to Kolmogorov, the complexity of an object is the length of the shortest computer program that can reproduce the object. Random objects, in his view, were their own shortest description. Whereas, periodic sequences have low Kolmogorov complexity, given by the length of the smallest repeating “template” sequence they contain. Kolmogorov’s notion of complexity is a measure of randomness, one that is closely related to Claude Shannon’s entropy rate of an information source.

I know that Our Masters loathe the very idea of excellence: “It’s so… inegalitarian! So… discriminatory! Now, ignorance, on the other hand, is something we can all share!”

But this is getting ridiculous! I mean, Kolmogorov actually served Stalin! He received the Order of Lenin! Almost certainly, he was a complete atheist! Doesn’t that count in the eyes of Our Dear Leaders?

Nope. Wrong race? Wrong sex? Down the memory hole you go!

Christians had better be smarter than our PC masters. A lot smarter, and a lot wiser.

As it is, Kolmogorov laid the foundation of probability theory, which has driven much 20th century math – which in turn has had a major enabler for our technological abilities.

Christians already have the theoretical underpinnings they need – not only the grand old Mathematics: Is God Silent? by James Nickel, but the recent Redeeming Mathematics: A God-Centered Approach by Vern S. Poythress.

Even a casual search will come up with all sorts of good material for math: most homeschoolers use Saxon Math, which gets the job done. (You’ll be extra satisfied with it, after Common Core is finished ravaging the student population…)

But it would be great if someone out there would develop material that is both inescapable Christian and absolutely excellent!

That is what I call God Honouring!