Mathematics is a thing apart, like nothing else in our world. It’s a human invention (I think), but not like other inventions. It exists in our minds and nowhere else – although its effects are felt everywhere since science, engineering, and technology have adopted it as their language. Yet, despite the fact that most people lump math and science together, they are very different.

Science is the observation of natural phenomena combined with the attempt to understand, explain, and predict what will happen next. It is rooted in the real world (and I’m going to avoid the whole metaphysical question about what is “really real.”) But although science uses mathematics, mathematics doesn’t use science. Indeed, it has been argued, and I’ve argued myself, that math is more akin to art or philosophy than it is to science, because at its center is an act of creation, often from nothing except the mind itself. Indeed, mathematics could not exist unless it were created by the human mind. I know from my own experience, and the descriptions of others, that arriving at a mathematical insight brings a sense of harmony, beauty, and wonder, which mathematicians call “elegance,” that is like the sense of creative rightness that artists experience. Indeed, one description of this creative sense, based on the observations of the 19th Century French mathematician, Henri Poincaré runs as follows:

Mathematical solutions are selected … on the basis of “mathematical beauty,” of the harmony of numbers and forms, of geometric elegance. “This is a true esthetic feeling which all mathematicians know,” Poincaré said, “but of which the profane are so ignorant as often to be tempted to smile.” But it is this harmony, this beauty, that is at the center of [mathematics].[1]

Why mathematics is different

And again, mathematics is unlike art or philosophy, because the creations of mathematics can be proven (within the assumptions that bound them) to be correct or incorrect. This is what it means to prove a theorem: you demonstrate its correctness. And there is no doubt involved; once proven, no one can later disprove such a theorem without altering the assumptions of the underlying mathematical system. No piece of art can be so demonstrated to be correct, and those works of philosophy that can be proven are, themselves, effectively sub-sets of mathematics based on logic.

I was struck by this conclusion in part because I recently read an article written in 1960 by physicist Eugene Wigner, called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.”[2] This classic essay describes the uncanny way in which natural phenomena can be described by mathematics, to the point where many natural phenomena, before they were understood, were correctly predicted by their mathematical descriptions. Wigner asks a central question: Why should this be? Why is it that an invention of the human mind can be capable of being so good at describing nature, particularly in cases when the mathematics involved was invented long before it was ever applied to science, as has happened in many instances in the history of science.

Wigner’s comments have been discussed and debated by many people since then. One commentator, physicist Max Tegner, suggested in 2007 that the reason that mathematics is so good at describing reality is that the physical world is completely mathematical. Others have commented that it is our own myopia that causes us to be able to explain the world in mathematical terms: we are only capable of observing those aspects of reality that are capable of being described by mathematics. Both statements are unprovable, and are regarded as a philosophical in nature, rather than mathematical, yet Tegner’s comment raises an interesting point that stimulated two additional thoughts.

Perhaps mathematics is reality

One is the slightly whimsical comment by nuclear physicists that subatomic particles are really just mathematical constructs. I suspect this is just a means of saying that subatomic particles are physicists’ way of balancing the books of quantum mechanics. But as someone trained as a mathematician, the thought struck me that perhaps this observation is literally true, that reality is nothing more than mathematics, and that explains why mathematics can describe reality, for they are the same thing.

The capstone of my conjecture comes from the cult-classic book that has been repeatedly ridiculed because it was a cult-classic, Zen and the Art of Motorcycle Maintenance. In this book, the author, Robert Pirsig, pursues the concept of Quality. To understand why this is important, you will have to read the book, which I recommend in any case. Pirsig is attempting to prove that the philosophic proposition that Quality is at the center of our perception of the universe, that, indeed, it creates the universe we perceive. In this, he is running counter to the large body of Western philosophic thought. Indeed, he had been challenged to prove that Quality existed, in part because he felt that it was vital to human existence, and that it was beyond our ability to define.

To prove the existence of Quality, he used a logical technique from the philosophical school of realism. His argument ran like this:

A thing exists if the world can’t function normally without it. If we can show that the world without Quality functions abnormally, then we have shown that it exists whether or not we can define it.

He then considered what our world would be like if we subtracted Quality from it. Art would cease to exist, because if you can’t distinguish good art from bad, then there is no point in art: a bare wall is as valuable as a painting (and let’s please agree to ignore so-called artists who produce bare walls and call it art). Likewise, poetry and literature would disappear, as would sports and acting and all kinds of performance. In the marketplace, only one kind of clothing would exist, and only plain foods. Indeed, virtually everything in our world would change, and dramatically for the worse, with one exception. In this way, he argued, it is demonstrated that Quality exists.[3] Now let me return to my own thoughts by drawing on his observation.

The one exception in Pirsig’s world-without-Quality was that mathematics, science, logic, and philosophy would be unchanged. He didn’t venture to guess why this should be so. I don’t know, either, but I would like to speculate.

Or perhaps mathematics contains reality

Perhaps mathematics is a thing apart from all the other things we create because it truly is the underlying reality of existence, the frame that contains existence. Perhaps it is mathematics that creates us, rather than the other way around, and it is our discovery of this underlying reality that leads to the mathematician’s sense of elegance and rightness.

I once speculated, in a presentation to a group of computer scientists, that once we learned enough about reality, we would find that reality was a computer (or rather, an information processor), that the elements of reality, from subatomic particles all the way up to ourselves and our minds, are packets of information, and that existence is the processing of these packets.

Or, to put it more poetically, perhaps we are merely mathematics in the mind of God.

It’s a thought.

[2] See, for instance, the following link for a copy of the original paper: http://www.physik.uni-wuerzburg.de/fileadmin/tp3/QM/wigner.pdf

[3] Pirsig, pp.193-4.

“What if I’m wrong? What else might happen instead?” This (two-part) question may just be the smartest question in the world, because it allows you to lift your head and look around at a broader world, outside the boundaries of your opinions and preconceptions. If you’re considering the future, as I do, then this is a critical question that can save your, uh, face repeatedly, and the current financial-crisis-and-recession environment is a perfect example.

This has been, and continues to be, an economic period for which there are no good precedents, and the only ones that come close, like the Great Depression of the 1930s, are unnerving, so that people don’t want to contemplate them. And there are so many things that can happen to change the course of events that trying to predict the One Right Thing That Will Happen is effectively impossible. But you still need to decide what to do, and that’s where this question becomes so valuable.

Strong Opinions, Weakly Held

So the first step is to come to a decision about what you think is going to happen. Someone once defined the job of a futurist as having strong opinions, weakly held. This means that, first, you have to have an opinion, but then, you have to be willing to change your mind when the facts indicate that you’re wrong. This goes against what most people do: they have weak opinions, strongly held, by which I mean that they usually don’t have strong opinions, but once they voice a thought, they will defend it to the death rather than admit they made a mistake, even in the face of clear evidence that they are wrong. This is exceptionally dangerous, especially in a world that changes as quickly as ours does.

In this case, I believe the current recession in America will continue throughout 2009, but with the signs and portents getting better and better as the year goes on. I expect the recession will end in early 2010, possibly as late as the 2nd quarter, but I doubt it will take that long. I also believe that growth will be so slow that unemployment will continue to expand, so that GDP and unemployment will both rise at the same time, which is both distressing and unusual. Moreover, I believe that people will continue to feel challenged by the economy through all of 2010 and into 2011. Towards the end of 2011, I expect things to be getting better, and people to be feeling more positive – and that’s the point where people will begin to believe the recession is over, even though it ended in 2009.

Meanwhile, the Rapidly Developing Countries (“RDCs”), notably China, India, and Brazil, will be growing more strongly, and recovering from their growth slow-downs more quickly. Their growth will put upward pressure on commodity prices, especially oil, faster than we would normally expect, so that inflation will re-emerge, particularly in energy prices and food, in 2010. If central banks don’t start to mop up all the excess liquidity in the global financial system at about that time, inflation could become a serious issue in 2011 and the years that follow.

Other Possible Futures

That’s what I think is most likely to happen. But what if I’m wrong? What might happen instead? Well, there are a number of alternative possibilities, also called scenarios, but let me focus on two in particular. The first is that there is another financial crisis. This could happen if a major, non-financial company runs out of cash, and slips into bankruptcy unexpectedly. They might be profitable, have significant assets and book value, but just run out of cash, and are unable to borrow working capital from the still shell-shocked banks. If that were to happen, it would panic the financial markets, because here is a solid, apparently stable company catching everyone off-guard with an unexpected bankruptcy. Or the banking system could go into another tailspin as loan losses, this time from commercial real estate, start hitting their balance sheets, triggering another round of unexpected bank runs. This, too, would trigger another panic. Or deflation, which is a vicious downward cycle of prices caused by weak economic demand, leading to weak prices, leading to weak profits, lead to weak demand, and so on. Any of these things could trigger another panic, and kill off any nascent economic recovery, producing a persistent recession or depression, which might last for years.

But I could also be unduly pessimistic about the economy. It might come back more rapidly than I expect, with scavengers swooping in to buy distressed houses at bargain prices, leading to a literal land rush in real estate, repairing bank balance sheets, and encouraging consumers. A stronger stock market, with rebounding values, would add fuel to the fire, and further increase confidence, leading consumers to start buying – cautiously and judiciously at first, then with greater strength and vigor. This could lead to an earlier recovery, stronger tax revenues, and a rapid rebound in government finances, especially at the state and local levels. In turn, stronger demand in the developed world would lead to a more rapid turnaround in the RDC economies, and trigger a faster rise in inflation, with oil pushing through $80 in 2010, and pointing much higher later. Indeed, if things get too strong, too quickly, we could see inflation return to levels unseen since the 1970s, and central banks being slow off the mark for fear of bringing the recovery to a screeching halt. This could lead to a new asset bubble, this time in commodities, with oil and food leading the way.

The Payoff

But however you see the future unfolding, the whole point in asking “What if I’m wrong?” is to give yourself greater breadth of vision, and allow yourself to make and, as appropriate, develop and use contingency plans to deal with alternative futures. There’s a structured way of doing this, called scenario planning – but that’s another rap for another time.

Oh, and one more advantage to asking “What if I’m wrong?”: it keeps you humble, and makes it easier to see the world as it really is, not as you expect it to be. And that may be the most valuable benefit of all.

Please feel free to pass this blog along to others you think might find it of value.

Best regards,
Richard Worzel