On Deconstructing Our Fortresses of Thought

“A country may be overrun by an armed host, but it is only conquered by the establishment of fortresses. Words are the fortresses of thought. They enable us to realize our dominion over what we have already overrun in thought.” - William Hamilton

“[W]e wrestle not against flesh and blood, but against principalities, against powers, against the rulers of the darkness of this world, against spiritual wickedness in high places... Stand therefore, having your loins girt about with truth, and having on the breastplate of righteousness;  and your feet shod with the preparation of the gospel of peace; above all, taking the shield of faith, wherewith ye shall be able to quench all the fiery darts of the wicked. And take the helmet of salvation, and the sword of the Spirit, which is the word of God.” - Ephesians 6:12-18

Hamilton’s metaphor characterizes the dominant approach to knowledge in the West. Our universities are structured to create a firm base of facts and knowledge about each field that the totality of human thought has been so neatly divided up and distributed down through the academic bureaucracy.

What Hamilton’s metaphor misses is that words don’t have to be fortresses. Better than fortresses are cultural exchange and mutual appreciation, which have grown out of economic relationships. These interactions keep the peace more effectively than fortresses, because they are flexible and mobile by nature rather than rigid and stationary.

While all our social institutions have forsaken flexibility in favor of rigid dogmatic procedures to varying degrees, I believe in mathematics, we find the tallest, most barricaded fortress. 

A Brief History of the Relationship Between Mathematics and Science

Everyone is familiar with Isaac Newton’s Theory of Gravity--the apple falls from the tree in the same way that the Earth “falls” in an orbit around the Sun, and that the moon “falls” in an orbit around the Earth. But have you ever considered how exactly gravity works? What’s the mechanism that causes the gravitational attraction between the Earth and the Sun, which is over 91 million miles away? How does the moon know that it’s supposed to keep circling around the earth, month after month after month? We know that the attraction increases proportionally to the mass of the objects, but why? If I want to keep a dog from running away from me, I need a leash or some device, at least. How does the Sun’s gravity pull the Earth onto its orbital path? Many scientists, from Isaac Newton to Stephen Hawking, have asked themselves this question, but there is still no definitive theory. Einstein’s Theory claims that “gravity is the curvature of space”, but this again just begs the question: “How does mass cause space to bend?” The leading mainstream theory--”gravitons”--still has no experimental backing whatsoever, and would explain very little anyway (Are gravitons a wave or a particle? etc.) We know that gravity “works” (well enough for measurements on Earth’s surface, at least), but we have never known how.  

So in science, while we can make accurate predictions about the real-world, in physics, chemistry, biology, etc., we rarely have a good explanation for why those predictions are true. 

Mathematics *should* be the opposite of this situation. 

Mathematics is the construction of symbols that in some way model reality, and we should be able to explain everything about the model, even if we still are clueless about reality. Knowing why everything works is what makes math interesting to me, and I believe, why its partnership with the sciences [see: Eric Temple Bell] has proven so successful over the past 400 years.

Tobias Dantzig writes a wonderful analogy explaining the nature of mathematics:

The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it. Then there is no end of surprise and of delight! - Numbers: The Language of Science. p. 240

No self-respecting mathematician would disagree with the idea that mathematics must, above all else, be explainable down to the level of axioms. But can this approach be taken too far? Can mathematicians become so obsessed with designing their garments that they disconnect from the world completely, such that none of their garments fit any more?

This is precisely what happened, culminating one hundred years ago in a famous [within mathematics, at least] controversy between two schools of thought: Formalism and Intuitionism.

Due primarily to Formalism’s victory in the controversy, most mathematicians today view math in precisely this way. Formalists focus on the development of language tools (algorithms), while disregarding the practical question of when and where they are useful. As a result we have such abstract theoretical fields as non-Euclidean geometry and topology, with no practical applications. 

David and Ellen Kaplan in their book The Art of the Infinite compare the Formalism of David Hilbert to Medieval theology:

The medieval view was that creatures-the created-glorify God; so if there were more creatures, then the greater would be the glorification. Hence if something could possibly exist, it would exist. The world-as crowded with beings as the Unicorn Tapestry-would then more loudly sing God’s praise. ln Hilbert’s terms this would translate to: since that which is consistent can exist, therefore it must. From this medieval standpoint, proving consistency would be enough to guarantee existence. Is it conceivable that Hilbert himself ever held this view? Could mathematical existence have meant this much-not this little-to him? - The Art of the Infinite, p. 51

Intuitionist mathematicians, on the other hand, view mathematics holistically within a broader social context. Mathematics is a technology, just like any other. We can greatly increase the number of “garment that fits” by identifying current problems in today’s world that can most benefit from the systematic approach that mathematics provides.  Intuitionists are more open to the search for space where our mathematical language-programs can usefully operate.

Why did Formalism win out? Ernst Snapper (Dartmouth College), in his essay on the [lack of a] philosophical foundation for mathematics, has this to say:

“These three reasons [Formalism being “more elegant”, classical proofs that Intuitionists reject, and Intuitionist proofs that are classically false] for the rejection of intuitionism by classical mathematicians are neither rational nor scientific. Nor are the pragmatic reasons, based on a conviction that classical mathematics is better for applications to physics or other sciences than is intuitionism. They are all emotional reasons, grounded in a deep sense as to what mathematics is all about. (If one of the readers knows of a truly scientific rejection of intuitionism, the author would be grateful to hear about it.)” - Mathematics Magazine, p. 212

The Formalists lost the intuitive knowledge of "mathematics as a technology". Even when directly confronted about this reliance and offered the viable alternative of Intuitionism, they still cling to the comfort elaborate language. (Perhaps mathematics is simply following the lead of physics.)

Where to from Here?

“One law for the Lion and Ox is Oppression.” - William Blake, The Marriage of Heaven and Hell

When it comes to learning, one size does not fit all. 
I grew up learning to play piano from reading the sheet music. I became proficient at sight reading piano music, and memorizing a piece by playing a measure or two over and over until my hands could play the measure automatically, out of habit. But despite my extensive training and practice, I have no ability to play a song by ear. I tried to play a measure of John Lennon’s Jealous Guy by listening to the piano part on Youtube, and I could not correctly hear the notes. All of my learning is mediated through the sheet music--my intuitive knowledge of playing piano is still that of a beginner.

When Elton John was 11 he could recite a 4 page Handel piece after hearing it just once. Imagine what the world would have missed out on if Elton John had been forced to learn to play from the sheet music instead of playing by ear. But that’s exactly how we are teaching mathematics today.

We need to teach student to “play math by ear.” What I mean is that students should learn mathematics intuitively, instead of a programmed instruction. At least, they should know that math can be done that way.

Our culture is obsessed with translating intuitive knowledge into a system of rules, but we don’t know how to translate it back the other way. We need to learn.

We need lesson plans that start from the rule-based knowledge of the Common Core Curriculum and create activities that stimulate intuitive knowledge for kids. To do this, I believe, teachers need to put passion, not necessarily into their relationships with students, but into the content that they teach. But isn’t it the students that bring the subject to life? I have to say “no.” I believe that subjects like mathematics and literature have a life of their own--not as a living organism, of course, but as the movements of culture.

G.W.F. Hegel famously wrote of Caesar:

“It was not, then, his private gain merely, but an unconscious impulse that occasioned the accomplishment of that for which the time was ripe. Such are all great historical men — whose own particular aims involve those large issues which are the will of the World-Spirit. They may be called Heroes, inasmuch as they have derived their purposes and their vocation, not from the calm, regular course of things, sanctioned by the existing order; but from a concealed fount — one which has not attained to phenomenal, present existence, — from that inner Spirit, still hidden beneath the surface, which, impinging on the outer world as on a shell, bursts it in pieces...
“Such individuals had no consciousness of the general Idea they were unfolding, while prosecuting those aims of theirs; on the contrary, they were practical, political men. But at the same time they were thinking men, who had an insight into the requirements of the time — what was ripe for development. … For that Spirit which had taken this fresh step in history is the inmost soul of all individuals; but in a state of unconsciousness which the great men in question aroused. Their fellows, therefore, follow these soul-leaders; for they feel the irresistible power of their own inner Spirit thus embodied.”

To create a historic change today, we need teachers who put thought into their work, and develop “insight into what is ripe for development” in technology and in our economy. Hegel explains here that historical change does not literally require individuals to “take the sword of the spirit” as St. Paul recommends in Ephesians. Hegel claims the requirements for change are thinking and insight. Whether St. Paul meant to limit his followers to a literal interpretation of the Gospel is a controversial topic. Perhaps “the sword of the spirit, which is the Word of God” means something much broader. It is such a broader “World-Spirit” that teachers must tap into in order that their students may develop an intuitive knowledge of the economic context and technological significance of the mathematical procedures they are being taught.