How Our Emotions and Bodies are Vital for Abstract Thought

Perfect Mathematics for Imperfect Minds

Anna Sverdlik

• Publisher: Routledge
• Year: 2018
• ISBN:
  9781351344753
• Edition: 1
• Topic: Psicología
• Subtopic: Historia y teoría en psicología

If mathematics is the purest form of knowledge, the perfect foundation of all the hard sciences, and a uniquely precise discipline, then how can the human brain, an imperfect and imprecise organ, process mathematical ideas? Is mathematics made up of eternal, universal truths? Or, as some have claimed, could mathematics simply be a human invention, a kind of tool or metaphor?

These questions are among the greatest enigmas of science and epistemology, discussed at length by mathematicians, physicians, and philosophers. But, curiously enough, neuroscientists have been absent in the debate, even though it is precisely the field of neuroscience—which studies the brain's mechanisms for thinking and reasoning—that ought to be at the very center of these discussions.

How our Emotions and Bodies are Vital for Abstract Thought explores the unique mechanisms of cooperation between the body, emotions, and the cortex, based on fundamental physical principles. It is these mechanisms that help us to overcome the limitations of our physiology and allow our imperfect, human brains to make transcendent mathematical discoveries.

This book is written for anyone who is interested in the nature of abstract thought, including mathematicians, physicists, computer scientists, psychologists, and psychiatrists.

1
THE UNREASONABLE EFFECTIVENESS OF MATHEMATICS

From Plato till now

In the fourth century BC, in his famous Dialogues, Plato expressed the idea that sacred truths exist outside of and apart from us, in some sort of world of their own, separate from ours: the world of pure “forms,” which later came to be called the Platonic World of Ideas. Plato posited that the Ideas that inhabit that realm are eternal, unchangeable, and absolutely real—more real than the constantly changing, mundane, and passing residents of the material world—or in other words, more real than you and I and everything our everyday understanding can grasp. Immortal souls, either not yet born in the flesh or already having departed from it, inhabit that World of Ideas and can come in direct contact with the Ideas themselves. But in the short interim between birth and death, when they are sent down to the material world, they completely lose touch with their prior experience. Only a few elect are sometimes able, in their lifetimes, to return there and regain their memory of certain aspects of the place. That is how the human intellect, though it rarely comes in contact with the world of ideas, can nevertheless manage to discern some of the laws of the universe. Our minds do not invent those laws, and do not create them using clever artificial constructions, but actually discover eternal concepts that have always existed, and always will: the absolute, objective kind of truth that does not depend on human perceptions.

This is one of the most profound and beautiful philosophical concepts of antiquity that is still current today. These days, the idea is especially widespread in physics and math. Roger Penrose, a prominent mathematician and physicist, for example, wrote the following on the subject:

I imagine that … when one “sees” a mathematical truth, one’s consciousness breaks through into this world of ideas, and makes direct contact with it … It is the essence of mathematical understanding … It is as though human thought is, instead, being guided towards some external truth—a truth which has a reality of its own, and which is revealed only partially to any one of us.

(Penrose, 2016, p. 554)

The main argument, and the most convincing one, in favor of mathematical Platonism appears in the title of a famous 1960 article by Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” especially in physics. The fact that math is indeed “unreasonably effective” is actually what makes the majority of mathematicians and physicists convinced Platonists.

Often we hear assertions to the effect that mathematics arose in response to the need to handle daily, practical tasks—count heads of cattle, measure and divide plots of land, and so on—and therefore speaking of it as a corpus of absolute truths outside of time and space is not an approach to be taken seriously. Some say that mathematical concepts cannot truly be discoveries; instead, they are invented and imagined by people, and they exist only inasmuch as the human reason that engendered them exists. In rebuttal, we can say that, first of all, the same thing is true of any other natural science, but that does not mean that the laws of, say, physics are the consequence of human needs or, to follow the argument to its absurd conclusions, that the American continent came to exist as a result of Columbus’s expedition, even though it also had thoroughly practical goals. In the second place—and this may be the more convincing counterargument—mathematical thinking, once it came about, very quickly extended far beyond our practical needs.

Here we are no longer talking about applied mathematics, although without it, of course, no precise discipline would have moved forward an inch; still, applied math, as a rule, does not play a key role in forming new hypotheses, the process at the heart of the hard sciences. Applied mathematics, in and of itself, does not produce scientific breakthroughs. But it does provide the groundwork for them, making them possible.

From here on, when we talk about mathematics, we mean theoretical mathematics, the type that long ago moved on past the requirements of cattle-counting and land-measuring, the kind that is born in the heads of occasional geniuses without any obvious connection with reality or with concrete, practical needs. It is more of a glass bead game, intellectual macramé, art for art’s sake. The results of this seemingly impractical exercise, over decades and centuries, often just gather dust in a corner, where they receive no attention from anyone not playing the same types of games. Then, one fine day, it is suddenly discovered that one of those useless ideas perfectly describes physical phenomena that its inventor never could have imagined, and we realize that the idea can provide a great deal of material good.

That inspiring scenario is not a one-off event. It has happened repeatedly throughout the history of science, especially in physics. Pythagoras himself noted that physical laws are subordinate to mathematical laws. He is said to have declared that all things are numbers, and that the universe is organized into a harmonious system of numbers and the relationships between them. Galileo Galilei, a founding father of classical physics, was also confident that mathematics was superior to the natural sciences, and said so quite categorically, when he stated that the laws of nature are “written in the language of mathematics.” Heinrich Hertz, a leading 19th-century physicist, made a similar assertion:

We cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than we originally put in to them.¹

And finally, one excerpt from that same article by Wigner may be more beloved than any other in contemporary math circles:

The mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena. This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language. (The emphasis is mine.—A.S.)

One of the most memorable, and most often cited, examples of what we have been talking about is the story of how Isaac Newton discovered the law of universal attraction. Newton proposed that there was a profound internal similarity between the curve described by a stone thrown on Earth and the trajectory of the moon’s orbit. Based on some very imprecise experiments and a single computational coincidence, Newton used the purely theoretical mathematical concept of the second derivative to formulate one of physics’ most fundamental laws. How Newton guessed that both of those trajectories formed types of conic sections, and why, from a huge number of other functions, he selected the second derivative, is a riddle, and the word “genius” does not explain it one bit—it only masks our own lack of understanding. One way or another, seizing on some very approximate experimental data and observations of an artificial, mental construction, he discovered a law that describes the motion of the heavenly bodies with a precision of greater than 99.999 percent.

Another example, just as popular, of the amazing correspondence between an abstract mathematical idea and the actual, physical world is the story of complex numbers. A complex number results from taking the square root of a negative number. That procedure, as any school math textbook will teach you, is impossible in principle and makes no sense at all, so in the past, complex numbers were also called “imaginary” numbers in every European language and my own native Russian as well. The idea of the complex number was born in the 16th century, when, of course, there was not the least bit of practical use for it. Instead, it was a sort of trick used to tackle Diophantine equations, which otherwise simply could not be solved. The results obtained turned out to be perfectly correct and immune to criticism, but the methods used to find them were worse than doubtful—they were simply unacceptable even from the mathematical point of view, not to mention opposed to common sense. Nevertheless, lacking other options, theorists were obliged to make their peace with the idea. With time, other areas of math, too, found it necessary to rely on this seemingly unnatural method of logical proof. Still, even though the ends justified the means, the chimera of the complex number continued to baffle the brain. Leibniz, for example, wrote that imaginary numbers were a “miracle of analysis, [a] marvel of the world of ideas, an almost amphibian object between Being and Non-being.”

But then three centuries passed, and quantum mechanics appeared, and it turned out that the concept of the imaginary number and mathematical models based on that concept are ideally suited to that field. Those same models are used today in fluid mechanics, geodesy, aeronautics, and many other applied sciences which provide us benefits that are far from imaginary.

Here is one more classical example, this time from the history of how elementary particles were discovered. In 1928, the great English physicist Paul Dirac, by manipulating abstract mathematical objects which had been conceptualized 15 years earlier and at first had nothing to do with physics or any other natural science, produced an equation that very precisely described the properties of the electron. The number of solutions to that equation was double the number of the known possible states of the electron. Among them were solutions that assumed the existence of a particle with a positive charge and a negative mass, with an absolute value equal to the mass of an electron. At the time, it was thought that only three elementary particles existed: the electron, the neutron, and the proton, and none of them had those characteristics. There was no experimental basis for assuming, back then, that there were more than three such particles, and the idea of a negative mass could only seem absurd.

Another person might have simply ignored that portion of the results, declaring them physically meaningless. That is a fairly common practice. For example, researchers routinely discard negative solutions they find when calculating distance, if the possible solutions to an equation include negative values along with positive ones (since a distance cannot be negative). But Dirac, who wrote later that “it is more important to have beauty in one’s equations than to have them fit experiment,” was not about to do that (Dirac, 1963). Instead, he proposed—exclusively on the basis of the formula he himself had come up with—that such a particle did in fact exist.

Dirac’s hypothesis was proven true in 1932, when a young American physicist named Carl David Anderson found experimental proof of the existence of the positron, the electron’s mirror-image twin, the first antiparticle known to science. (The expression “found proof,” by the way, is somewhat of a stretch in this case. Anderson, as far as we know, was not even aware of Dirac’s hypothesis about antimatter.)

Those three examples (and there are many more of them, too) should provide enough impetus to agree with Pythagoras, Galileo, Hertz, and Wigner: Mathematical formulas are not artificial human inventions. They are the language of creation. We do not invent that language. We only gradually learn to understand it.

* * *

What exactly is the Platonic World of Ideas? What is it like? No scientific evidence of its existence has been found, of course, but do we need any? Is evidence even possible? Plato himself asserted that the place was absolutely real, but what did he mean by that, and what do contemporary mathematical Platonists mean when they say it? Certainly, nobody today imagines the world of forms as akin to a sort of International Bureau of Weights and Measures, where the walls are decorated with mathematical templates, perfect cubes and spheres, straight lines of zero thickness, and infinite quantities to behold at a glance. Probably even Plato did not have that in mind. It is completely possible that the world of Platonic ideas is in fact a pure and unadulterated idea itself, a philosophical concept, and while it is correct, it is unprovable in principle, as unprovable in principle as any other profound philosophical concept or the existence of God. (After all, we cannot deny the existence of God solely on the basis that nobody has ever seen Him.)

On the other hand, it could be that this is just one of our collective fantasies. All peoples of the world have legends about something like a fount of wisdom or a storehouse of all knowledge. But nobody much believes those tales past the age of six or so. What is it, in principle, that makes the world of Platonic ideas so different from other, similar archetypes? Yes, it has no concrete features, and it is unattainable, which makes it all the more attractive. And we learned about this world from the great Plato, not just any old philosopher, which also tends to make a serious impression on intellectuals. But in this case, the facts and evidence so cherished by scientists are of secondary importance, and all discussion of the concept, even when the people involved are men of science, seems more theological than scientific.

Somehow or other, the world of Platonic ideas, enchanting and radiant, divulging its secrets only to the chosen few, in and of itself seems ultimately cut off from our powers of understanding. For some, it is the essential, objective truth, but it is a truth that exists independently of human beings and is inaccessible to detailed human thought; for others, it is nothing more than the fruit of our imaginations, a beautiful illusion that lives only in our heads.

I believe that the situation is not as hopeless as it seems, and that there are elements of this question that can still be explained. That explanation should also help to bring closer together positions that at first glance seem directly opposed to one another. No evidence of a theological nature will be necessary. In the overarching question of how we learn about the world, after all, there are two obvious players at work—we people and the rest of world—and both those things are completely real. So let us assume that the question of how our brains draw information from the outside, and how our brains transform that information and what they transform it into, is an ordinary question for the natural sciences. And if we can explain that, if we can show that we do not need any sort of third-party intermediary in order to conceptualize the fundamental laws of nature, then that means that the metaphysical world of Platonic ideas really does only exist ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• List of figures
• Preface
• 1. The unreasonable effectiveness of mathematics
• 2. Why logic is never ideal
• 3. Working memory and logical limitations
• 4. Overpowered by emotion
• 5. From cognition to recognition and back again
• 6. Non-algorithmic thinking machine?
• 7. How mathematics can outwit physiology
• Afterword
• Index