How Did Einstein Think?

John D. Norton
Department of History and Philosophy of Science
and Center for Philosophy of Science
University of Pittsburgh
http://www.pitt.edu/~jdnorton

This page at http://www.pitt.edu/~jdnorton/Goodies

Prepared to accompany
cityLIVE! "everything einstein"
November 15, 2007, 6:30 pm
The New Hazlett Theater, Pittsburgh PA
Panelists: John D. Norton, Walter Isaacson
Moderator: Regina Schulte-Ladbeck

What is it about Einstein that we find so fascinating? One side is that his thought and achievement seems so far above our mundane sphere that we are inspired merely reflecting on him. There is a second side to this fascination. It lies, I believe, in the thought that we are in the end not so far removed from him. While his achievements are transcendent, he was in the end simply human. He ate soup and paid taxes like the rest of us. He tried to keep ahead of the mundane practicalities of life before they submerged him. He didn't wear socks and thought just one soap for washing and shaving was quite enough.

It is that second sense that has always fascinated me. What Einstein did, he did using tools available to all of us. He had no magic wand or secret subscription to Encyclopedia Galactica, where all the truths of nature are written. He used tools and methods available to everyone. He read the same text books and journals available to every scientist of his day. His principal tool was a notepad with a pen and pencil. He read and wrote and calculated and thought; and out poured his extraordinary achievements.

In this regard, Einstein is part of an extraordinary tradition of achievement in science that extends to antiquity. I am equally fascinated by the ancient astronomers. Look around you. How could you figure out that the surface of the Earth is spherical--not just curved, but a surface of constant curvature in all directions? The ancient astronomers did it with nothing more than patient observation, pointed sticks and ingenious thought. Well, perhaps I am a little more fascinated with Einstein than the ancient astronomers.

My purpose here is to say something about how Einstein worked and thought. I've written a lot elsewhere on the details of Einstein's discoveries. Now I want to try to get behind them and understand better the thinking that led Einstein to them.

Einstein's Answer

Perhaps we should heed Einstein's advice:

"If you want to find out anything from the theoretical physicist about the methods they use, I advise you to stick closely to one principle: don't listen to their words, fix your attention on their deeds."
"On the Methods of Theoretical Physics," Herbert Spenser Lecture, Oxford, June 10, 1933.

What can we learn about Einstein's thoughts and methods by looking at his work? I'll sharpen my queries in a series of questions that span from the general and big to the narrow and specific.

Inscrutable Flash of Insight
or
Systematic exploration?

Let us start with the biggest question first. When Einstein made his big discoveries, how did they happen? Did they come in momentary flashes of insight, the product of hidden processes that suddenly flood the mind with their perfected insight? Or did they come through slow labor, much as we might build a tall tower by patiently and systematically piling the bricks one upon another?

Einstein's major discoveries seem to have both in them. Let us consider two discoveries.

Unfettered Speculation
or
Controlled Flight?

Einstein did have decisive flashes of inspiration. What was their character? Were they just wild moments of unfettered speculation? Or were they controlled? The first fits with the image of Einstein's scientific creativity as resulting from a liberation from reason. The second fits with it as a more profound fulfillment of reason.

My view is that his flights fit much better with the second. In the two cases we have just seen, the moment of inspiration wasseeded by earlier work and directly responded to it.

The insight over the relativity of simultaneity came after years of struggle with failed efforts to reconcile the principle of relativity with Maxwell's electrodynamics. He did not so much choose to leap into the unknown as he was pushed by the accumulated pressure of those problems.

Things were similar but not so acute with the "happiest thought of [his] life," the principle of equivalence. It emerged after a series of failed attempts at finding a relativistic theory of gravitation. All the candidates he thought up failed to fit a simple expectation about gravity that he traced back to Galileo: all things big and small must fall with the same acceleration. The virtue of the gravitational field created by acceleration was that it was assured to have exactly this property.

Thinking Mathematically
or
Thinking Physically?

What sort of theorist was Einstein? Everyone knows that he brought profound insight in the form of new theories in mathematical physics. If we seek the origin of that thought, should we put the emphasis on the "mathematical" or the "physical"? The distinction between the two is one that Einstein himself drew and reflected upon.

In thinking mathematically, or, as Einstein's sometimes said, formally, one takes the mathematical equations of the theory as a starting point. The hope is that by writing down the simplest mathematical equations that are applicable to the physical system at hand, one arrives at the true laws. The idea is that mathematics has its own inner intelligence, so that once the right mathematics is found, the physical problems melt away. Philosophers will recognize this as a form of Platonism.

In thinking physically, one proceeds quite differently. The starting point is physical. Perhaps it lies in an examination of the results of experiments; or in an insistence on certain founding physical principles, like the principle of relativity of the conservation of energy and momentum; or perhaps it draws on deeper, more visceral ideas of what can and cannot happen in real physical situations.

Einstein's theories sometimes made special mathematical demands on physicists. His great achievement, the general theory of relativity, required physicists to learn what we now call "tensor calculus," which proved something many found formidable. But it was not mathematics that drove Einstein's theoretical successes.

Einstein's enduring achievements in physics were virtually all a product of the earlier part of his life: special relativity, Brownian motion that shows the reality of atoms, the light quantum, general relativity, the "A and B" coefficients paper that grounds lasers. All this was done before he turned 40. Virtually all of these achievements depended upon a very astute form of physical thinking. Einstein then dressed it in mathematical clothing, seeking where ever possible to keep the mathematics as simple as he could.

Just how did Einstein's physical insight work? One part was an keen instinct as to which among the flood of experimental reports were truly revealing. Another was his masterful use of thought experiments. Through them Einstein could cut away the distracting clutter and lay bare a core physical insight in profoundly simple and powerfully convincing form.

Here's an example of a thought experiment from Einstein's work on general relativity already mentioned above. Einstein realized at the start that we needed to think differently about the empty space of special relativity if we were to arrive at an acceptable relativistic theory of gravity. We needed to stop thinking of it as the gravitation free case. It is actually the simplest case of a gravitational spacetime. His claim, the principle of equivalence, was that uniform acceleration in empty space produces a gravitational field.

But how can that be made intelligible? Einstein's approach was simple and ingenious. He had us to imagine a physicist who is drugged and placed inside a box. That box is transported to a distant part of space where it is accelerated uniformly by some agent. The phyicist wakes up. Any object released by the physicist would accelerate to one part of the box. The acceleration would be the same for all objects released, be they small or large in mass. That is the distinctive property of gravity: it accelerates all masses alike. Could the physicist know that the box is accelerated unformly in space and not at rest in a homogenous gravitational field? No, Einstein asserted, there is no way. There is no difference between the two cases. The result of the acceleration is to create a gravitational field inside the box.

Geometric
or
Algebraic Thinking?

Einstein's work continued to depend on mathematics and the mathematics he employed grew more as time passed. There are different ways of using mathematics to clothe a theory and Einstein adopted a particular stance towards them. The two ways I will describe will be recognized by anyone who has had a high school exposure to mathematics.

Another approach is algebraic. This is a way of doing mathematics that concentrates on writing symbolic expressions and manipulating them. "A fish is 10 inches long plus half of itself. How long is the fish?" We say. "Let x be the length of the fish. Then x = 10 + x/2. Solving, we find x=20. The fish is 20 inches long."

Here's another problem. Take the formula y=x2. Is it changed when we replace x by -x? A quick calculation shows that it is not. y=x2 becomes y=(-x)2 which is just y=x2 again since (-x)2 = x2.

This last problem is purely one of symbolic algebraic manipulation. Yet it is really just the same the problem as the symmetry of the parabola. If x and y are ordinary Cartesian coordinates the y=x2 is the formula for a parabola. Replacing x by -x is just the operation of reflection over the y axis. So when the formula y=x2 stays the same under this transformation, we have the algebraic equivalent of the demonstration of the mirror symmetry of the parabola.

We see the difference of approach expressed directly in physics. Consider special relativity. Most treatments of the theory rapidly seek to develop the notion of spacetime. Through it one learns a wonderfully simple way to conceive the theory. All of space and time taken together form a single spacetime. Special relativity is really just the theory of this spacetime's geometry. One uses the theory by drawing straight lines and hyperbolas and forming geometric constructions similar to those we learned in elementary geometry classes. The principle of relativity is expressed as a sort of isotropy of the spacetime, somewhat akin the isotropy of a Euclidean space--that all its directions are the same.

That geometrical way of conceiving special relativity is not Einstein's. It was devised by the mathematician Hermann Minkowski shortly after Einstein published his special theory of relativity. Einstein was reluctant to adopt Minkowski's method, thinking it smacked of "superfluous learnedness." It was only well after many others had adopted Minkowski's methods that Einstein capitulated and began to use them. It was a good choice. It proved to be an essential step on the road to general relativity.

Einstein preferred to think of his theory in terms of the coordinates of space and time: x, y, z and t. The essential ideas of the theory were conveyed by the algebraic properties of these quantities, treated as variables in equations. Its basic equations are the Lorentz transformation, which, in Einstein's hands, is a rule for changing the variables used to describe the physical system at hand.

The laws of physics are written as symbolic formulae that include these coordinate variables. The principle of relativity of relativity then became for Einstein an assertion about the algebraic properties of these formulae; that is, the formulae stay the same whenever we carry out the symbolic manipulation of change of variables of the Lorentz transformation.

The emphasis in Einstein's algebraic approach is on variables, not spacetime coordinates, and formulae written using those variable, not geometrical figures in spacetime.

For many purposes, it makes no difference which approach one uses, geometric or algebraic. Sometimes one is more useful or simpler than the other. Very often, both approaches lead us to make exactly the same calculations. We just talk a little differently about them.

However there can be a big difference if we disagree over which approach is more fundamental. We now tend to think of the geometric conception as the more fundamental one and that Einstein's algebraic formulae are merely convenient instruments for getting to the geometrical properties.

There is some evidence that Einstein saw things the other way round. He understood the geometric conception, but took the algebraic formulation to be more fundamental. A simple example illustrates how this difference can matter a lot.

One could take two attitudes to this little oddness. One could think geometrically. One could say that what we are really talking about are just straight lines in Euclidean space. So the limit of the sequence is just a vertical line. What has gone wrong is that algebraic device for representing lines as formulae has "gone bad" in this limiting case. So we shrug the whole thing off as an inadequacy in the algebraic representation.

One might also think algebraically and take the formulae to be fundamental. Then one would say that the set of structures represented themselves go bad in the limit. That is, they are "singular" to use the fancier term when infinities like this appear. That amounts to saying that something odd is afoot as we approach the limit of the sequence.

One arose in the later 1910s shortly after the birth of relativistic cosmology. Einstein became engaged in a dispute with de Sitter and others over a new cosmological spacetime explored by de Sitter.

Thinking geometrically, we would just say that there is no pathology on this surface. It is just that our coordinates used to describe the surface at those points are not working well any more. How could it be otherwise, we might wonder, since all points in the spacetime are the same geometrically.

For Einstein, the algebra trumped the geometry and he found pathologies where we now think there are none.

In Sum

The transcendent achievement of Einstein required many components. He needed an intellect with singular powers. He needed a dedication to hard work. And he needed a commitment to finding the right answer, no matter how hard the path became.

We can learn just a little more from this glimpse of how Einstein thought. For Einstein had quite particular strengths. They lay in an acute physical intuition that guided him to the fertile physical ideas and revealing experimental results. As long as Einstein used those particular skills to drive his work, he produced one great success after another. He was the right thinker, with the right abilities, in the right place and time.

Then Einstein changed his thinking. He was no longer guided so much by physical intuition as by mathematical simplicity. As that change set in, Einstein's work began to languish. He retreated more and more into a private world in which he spent decades searching for his unified field theory. While he did that, the mainstream of physics moved on to elaborate the quantum theory. Einstein did not follow. His physical instincts told him that this was not the fundamental theory; that was to be found by the mathematical path.

We might lament that Einstein's work took this pathway. What might have happened had he continued to follow his older methods? We cannot know. However I suspect that not much would have emerged. Einstein's physical instincts were the ones needed to develop relativity theory and his other successes.

When the focus of research moved to quantum theory, a different sort of instinct seemed to be required. That was embodied by the Danish physicist Niels Bohr. He had a characteristic tolerance and even delight in contradiction. That characteristic enabled Bohr to theorize successfully in the bewildering and uncertain quantum domain and in a way that Einstein's physical sensibilities found repugnant. Einstein's role changed to that of a senior sage, warning the new generations of the dangers of the path they had chosen.