Did Einstein Not Understand Math?



What can you expect from a mathematical theory developed by someone who did not understand mathematics?

Princeton is a madhouse and Einstein is completely cuckoo. (Robert Oppenheimer January 1935

Use Your Brains!

How are we going to choose between fact and fiction, between science, pseudo-science and nonsense in the flood of information overwhelming us every day? Well, use your brains! What else?

Let’s test this approach on the foundation of modern physics:

                                                     The break-through in the US in 1919.

Nobody Understands, Nobody Even Claims to Understand

But relativity theory is very difficult, you say,  and my brains are not sharp enough to make it possible to say anything sensible about this subject. Don’t panic, you are in good company, because not even expert physicists claim to understand:
  • Many people probably felt relieved when told that the true nature of the world could not be understood except by Einstein and a few other geniuses who were able to think in four dimensions. They had tried to understand science, but now it was evident that science was something to believe in, not something which should be understood. (Hannes Alfven Physics Nobel Laureate 1970)
  • The relativity of space and time is a startling conclusion. I have known it for more than 25 year, but even so, whenever I quietly sit and think it through, I am amazed… It is not the depth of mathematics that makes Einstein’s relativity challenging. It is the degree to which the ideas are foreign and apparently inconsistent with our everyday experience. (Brian Greene in the Fabric of Cosmos) 

  • People are applauding me because everybody understands what I say, and you because nobodunderstands what you say. (Chaplin to Einstein) 
  • Why is it that nobody understands me, and everybody likes me? (Einstein in New York Times 1944)
  • ... the general theory of relativity. The name is repellent. Relativity? I have never been able to understand what the word means in this connection. I used to think that this was my fault, some flaw of my intelligence, but it is now apparent that nobody ever understood it, probably not even Einstein himself. (Synge) 
  • I myself can hardly understand Laue’s book. (Einstein about the first text book on special relativity 1914) 
  • Thus we can sum up: general relativity can not be physical, and physical relativity is not general. (Fock)
  • Scientific American once ran a competition offering several thousand dollars for the best explanation of Einstein’s general theory of relativity in three thousand words. “I am the only one in my entire circle of friends who is not entering,” Einstein ruefully remarked. “I don’t believe I could do it”.
  • Einstein, my upset stomach hates your theory [of General Relativity]—it almost hates you yourself! How am I to’ provide for my students? What am I to answer to the philosophers?!! (Paul Ehrenfest)
  • The general public looked upon relativity as indicative of the seemingly incomprehensible modern era, educated scientists despaired of ever understanding what Einstein had done, and political ideologues used the new theory to exploit public fears and anxieties—all of which opened a rift between science and the broader culture that continues to expand today. (David Cassidy)

                                                        I don’t understand. Do you?

OK fine, nobody understands relativity, not even Einstein, but how are we then going to judge? There are two possibilities:
  • (A) Since nobody understands, the theory is most likely correct and meaningful.
  • (B) Since nobody understands, the theory does not make sense.

(A) or (B)? To seek an answer, let us take a look at the inventor Einstein himself. Maybe he can give a clue. Like most of us, Einstein had difficulties with mathematics:

  • Einstein was a lazy dog. (Minkowski, Einstein’s math teacher)
  • neglected mathematics…because my intuition was not strong enough to differentiate the fundamentally important from the dispensable erudition…(Einstein)
  • Do not worry about your difficulties in Mathematics. I can assure you mine are still greater. (Einstein to Chaplin/Marilyn Monroe)

You can find further inspiration on Einstein: Stupid, Unhappy and Lonely.

                                                                    The Genius

Can Mathematics Really Do Wonders?

So Einstein did not understand mathematics, but like most people sharing this experience, Einstein had a very high opinion of the value of mathematics:
  • The supreme task of the physicist is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction. 
  • It is marvelous that man is capable of reaching such a degree of certainty and purity in pure thinking as the Greeks show for the first time to be possible in geometry. 
  • Pure mathematics is, in its way, the poetry of logical ideas.
  • If the facts don’t fit the theory, change the facts.
  • One reason why mathematics enjoys special esteem, above all other sciences, is that its laws are absolutely certain and indisputable, while those of other sciences are to some extent debatable and in constant danger of being overthrown by newly disco.
  • Mathematics are well and good but nature keeps dragging us around by the nose.
  • But there is another reason for the high repute of mathematics: it is mathematics that offers the exact natural sciences a certain measure of security which, without mathematics, they could not attain.
  • But the creative principle resides in mathematics. In a certain sense, therefore, I hold true that pure thought can grasp reality, as the ancients dreamed. 
  • How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom.
  • Yes, we have to divide up our time like that, between our politics and our equations. But to me our equations are far more important, for politics are only a matter of present concern. A mathematical equation stands forever.

                                                    The Musician and the Sailor
Relativity theory is a mathematical theory. What can we expect from a mathematical theory developed by someone not understanding mathematics? Someone who cannnot “differentiate the fundamentally important from the dispensable erudition…”
  • (A) It is likely to be correct and meaningful.
  • (B) It is likely to be non-sense.

(A) or (B)? Maybe Einstein himself gave us a clue:

  • Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.
  • It strikes me as unfair, and even bad taste, to select a few individuals for boundless admiration, attributing superhuman powers of mind and character to them. This has been my fate, and the contrast between the popular assessment of my powers and achievements and the reality is grotesque.
  • I don’t believe in mathematics.
  • A question that sometimes drives me hazy: am I or are the others crazy?
  • When I examine myself and my methods of thought, I come to the conclusion that the gift of fantasy has meant more to me than any talent for abstract, positive thinking.
  • Here in Princeton I am considered an old fool.
  • What I wanted to say was just this: In the present circumstances, the only profession I would choose would be one where earning a living had nothing to do with the search for knowledge. (Einstein’s last letter to Born)

The Nobel Prize Motivation

In any case the Swedish Nobel Committee was not convinced:

  • The ROYAL SWEDISH ACADEMY of SCIENCES has at its assembly held on November 9, 1921, an accordance with the stipulation in the will and testament of Alfred Nobel, decided to independent of the value that (after eventual confirmation) may be credited to the relativity and gravitational theory bestow the prize that of 1921 is awarded to the person in the field of physics who has made the most important discovery or invention to Albert Einstein for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect.
This is the only Prize motivation expressing explicitely for which contribution the Prize is NOT awarded: Relativity!


Einstein liked contradictions and ambiguity:

  • There is no aether according to the special theory of relativity (1905)
  • There is an aether according to the general theory of relativity (1920)
  • The scientist must appear to the systematic epistemologist as a type of unscrupulous opportunist: he appears as realist insofar as he seeks to  describe a world independent of the acts of perception; as idealist in-sofar as he looks upon the concepts and theories as the free inventions of the human spirit (not logical ly derivable from what is empirically given); 
  • as positivist insofar as he considers his concept and theories justified only to the extent to which they furnish a logical representation of relations among sensory experience. He may even be viewed as Platonist or Pythagorean insofar as he considers the viewpoint of logical simplicity as an indispensable and effective tool of his research.

Further Reading and Conclusion?

If you feel you want more information, take a look at Many-Minds Relativity and Mathematics = Magics? 

You may also ask a physicist/mathematician if he/she really understands relativity theory, and not just pretends to understand. For example, ask for and explanation of curved spacetime.

You can also ask if the constancy of the speed of light in vacuum is a definition or a physical fact, see Is One Dollar = One Euro. 

After this, you might be ready for a conclusion, which you are wellcome to post below. (A) or (B)? 

Or is it so that we as more or less educated citizens in a more or less democratic society, are not expected to have any opinion, about relativity or other technical matters? Should we simply leave relativity theory to the physicists and technical matters to the specialists, even if they don’t seem to understand, and just pay our tax bills to science and keep quiet?

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