Naming Infinity - maths and spirituality

There is a book that's still hot off the press, called:

Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity

The authors, Graham and Kantor, tell the story of the rise of the famous Russian school of mathematics, and it's intimate relationship with an Orthodox movement (later deemed heretical) called 'name worshipping'.
One of the main interesting features about this work is the way in which it illustrates, by way of historical description, how mathematics and religious spirituality can still, thousands of years after Pythagoras, inform each other in a practically efficient and fruitful way.

Russell said that mathematics is the chief source of the belief in exact and eternal truth. Of course he had a priori, due to his other beliefs, discarded another important source of such belief: religion, or more generally, spirituality. What is the unerlying reason why these two areas of human endeavor keep popping up hand in hand throughout human history? Howcome they are so seemingly different, yet clearly intimately connected? One answer is to explain everything in terms of human brain function and biological evolution. However, these kinds of explanations fall short of explaining the unchangeing characters (Russell used the word 'eternal') of these subjects. This, and other drawbacks of the "all in the mind" theory, is something I will return to in this blog.
Another explanation is given by a model which includes non-empirical and unchangeing objects which constitute the 'forms' of the physical, inherently dynamical and changeing, entities we observe empirically.

Freedom is to be able to say the Truth

In George Orwell's famous novel Nineteen Eighty-Four, the protagonist Winston Smith writes:

"Freedom is the freedom to say that two plus two makes four. If that is granted, all else follows."

The phrase "two plus two equals five" was used in communist USSR to refer to how five-year plans were supposed to increase production to yield more than previous methods of production. Everybody knows what this denial of truth led to. However, this scenario is not something exclusive to so-called totalitarian regimes. It also has other manifestations:

• The current financial crisis might at one level have its foundation in the (tacit) belief that money can grow on its own - that one can somehow get more than there actually is.
  
• Relativism is very popular in today's Western world, and ideas of social construction permeate contemporary thinking. In some parts of this view, science is just useful conventions and has no reality outside of human existence.
  

As Aldous Huxley anticipated already in the 1930s, the dictatorship is not likely to come the way George Orwell thought, that is, via forceful external pressure, but it will gradually grow into our lifestyle in ways which we welcome, because we find them pleasurable. However, in contrast to what both Orwell and Huxley thought, the dictatorship does not seem to be a regime. It's not a government who wants to control the citizens. It's our very culture which is going to force us into dead-end roads, like that of relativism and individualism.
As the examples above show, the way this can happen will most likely go via a rejection of objective truth. In this regard, the ultimate triumph is to overtake the main bastion of objective truth: mathematics.
To quote again from Nineteen Eighty-Four:

"In the end the Party would announce that two and two made five, and you would have to believe it. It was inevitable that they should make that claim sooner or later: the logic of their position demanded it. Not merely the validity of experience, but the very existence of external reality, was tacitly denied by their philosophy. The heresy of heresies was common sense. And what was terrifying was not that they would kill you for thinking otherwise, but that they might be right. For, after all, how do we know that two and two make four? Or that the force of gravity works? Or that the past is unchangeable? If both the past and the external world exist only in the mind, and if the mind itself is controllable—what then?"

Contemporary relativists think that they are pursuing a liberating ideology. It is more likely, given the evidence, that the result will be the exact opposite. Objective truth has always been a defender of freedom, and mathematics is its most clear and universal manifestation. Freedom is to be able to say that mathematics is true and unchangeable no matter what our brains are like, and no matter what our culture and politics says.

Relative plausability of platonism compared to standard hypotheses

Science shows that reality is so rich and complex that explanations will inevitably lead to scenarios which from our every day perspective seem fantastic, counter intuitive, and beyond common sense. Philosophy on the other hand, uses as one of its most widely applied tools of argumentation, the principle that a statement, in order to be believable, must be readily understandable, at hand, and well-grounded in the prevailing ontology (nowadays most often materialism). Science provides many examples that this principle is inadequate in our pursuit of understanding of the world. Indeed, there exist examples where scientific reasoning leads us to quite fantastic scenarios as the best explanations of the world.

Modern theoretical physics, in considering the anthropic principle, has brought us to basically two plausible scenarios: Either there is a Creator who has fine tuned the physics in our universe so that to make life like ours possible, or there is an incredibly high number of universes besides ours, where all logically possible physical theories are realized.
Now, which of the following would you say requires the greatest leap of faith, mathematical platonism, which posits a realm of mathematical objects and relations existing independently of our own minds, or the existence of God or infinitely many universes outside of our own?
If string theory with its ten spacial dimensions or the multiverse with billions (or even infinitely many) different universes outside our own are the best explanations for how physical reality is, and if it is reasonable for a rational person to believe in such theories, then what is so strange about the belief that mathematical objects exist independently of our minds?

It is a simple task to realize that both God and the structure of masses and masses of different universes are more complicated structures than the realm of mathematics. Still it is rational to believe in one of these extremely complicated transcendental existences. In comparison, a platonic realm of mathematics, seems like a small grain of sand.

The above examples show that the philosophical argument against mathematical platonism based on the difficulty to imagine where a transcendental world like the platonic realm could exist, is an argument which is not universally applicable. It appeals to down-to-earth bias (one could say geocentrism, to resurrect an old piece of terminology in a new context), but there exist situations where it is misguided, for example in the multiverse scenario mentioned above. To realize the shortcomings of the geocentric argument, one has to take a huge leap into abstraction. To cling to the argument is to effectively say "I can't make the required leap of abstraction, therefore I don't believe in the possibility of those things".

The most vocal current anti-platonists do not reject platonism mainly on account of the geocentric argument, but because they are in the business of promoting their own alternative views which underlie their research agendas. Did you expect anything else?

Musical platonism

Mozart is the greatest composer of all. Beethoven created his music, but the music of Mozart is of such purity and beauty that one feels he merely found it — that it has always existed as part of the inner beauty of the universe waiting to be revealed.

- Albert Einstein



Mozart's music is the mysterious language of a distant spiritual kingdom, whose marvelous accents echo in our inner being and arouse a higher, intensive life.

- E.T.A. Hoffmann

New Scientist article about mathematics and computing

A recent article in New Scientist

http://www.newscientist.com/article/mg20126971.800-rise-of-the-robogeeks.html

describes, in the typical vague and sensationalist terms characteristic of so-called popular science journalism, some work of computer scientists Aaron Sloman and Alison Pease, respectively.
After the initial spectacular suggestion that artificial mathematicians may become a reality, the article eventually admits that the actual aim of the research presented is much more modest:

"at this stage he [Sloman] is simply trying to show a link between spatial manipulations and the basics of mathematics."

One should be very careful in extrapolating this into believing fantastic stories about "robot mathematicians". It is also important to remember that the idea of mathematical machines that could prove theorems more or less independently of human input, is not a new one. What can be done today to some extent, is automatic proof check. Several mathematicians have constructed programs that can formally check the validity of certain non-trivial theorems in mathematics, that is, the program finds a deduction from the axioms to the full theorem, using only formal rules of logic. It is quite possible that similar methods could be used to find previously unknown proofs of mathematical results or conjectures, or to handle routine arguments that mathematicians today have to do by hand.
However, mathematics does not simply consist of formal truths deducted from logic, just as a painting does not simply consist of spots of paint on a canvas. Many formal truths are not relevant, and human mathematicians can easily exclude those, but it is very hard to imagine a computer program which would recognize this unformalized (unformalizable?) difference. For instance, a computer might come up with the theorem 2 + 2 = 4, which is relevant, but it could just as well come up with the equivalent theorem "the even prime number plus itself equals the smallest positive even integer square", which is not a mathematically relevant theorem. How is the computer to know which one to choose? It seems it would need human input.
It seems unlikely that a computer, which necessarily works with a formal system as a model of mathematics, will, from some rudimentary axioms alone, come up with such creative products of the human mind as integrals, the category of groups, or etale cohomology, concepts of central importance in modern mathematics. This is because although the proving of theorems seems to be formalizable, the formation of good mathematical definitions, that is, fruitful and enlightening concepts, is something that seems to require human judgement.

The black-and-white statement by Pease that mathematics is not discovered but created is just an empty slogan unless one provides some good argument for it. It is also something that many professional mathematicians would object to. (Indeed, comments at the New Scientist website by readers of the article react against precisely this statement).
It is striking that an article in New Scientist about the possibility of artificial mathematicians, do not feature any statement by an actual mathematician. If you want the best possible opinion about your health, you ask a medically trained person. If you want the most accurate description of history, you ask a historian. If you want the best insight into mathematics you ask a...computer scientist or cognitive psychologist? This reminds me of a quotation by A. N. Whitehead:

"Philosophers, when they have possessed a thorough knowledge of mathematics, have been among those who have enriched the science with some of its best ideas. On the other hand, it must be said that, with hardly an exception, all remarks on mathematics made by those philosophers who have possessed but a slight or hasty and late-acquired knowledge of it, are entirely worthless, being either trivial or wrong."

Chartres cathedral

Here I am standing
as if I was here already a thousand years ago
at this work of human hands
a gift to infinity and grace