Hi.

If you ask any physicist, she will tell you that Einstein is, without doubt, among the top 5 physicists of all time; if not the greatest. He will be there with Newton and three more.

However, if you ask a random mathematician, Godel, even though a true genius, will certainly not rank so high. Maybe in the top 10? Gauss, Euler, Newton … mmm … I gave my doubts.

So, I don't think, and most physicists and mathematicians will agree with me on this, that Godel is the Einstein of mathematics.

Hope it helps. For more, see my profile.

Hmm… okay :-)

So, you think that belief in a personal God is illogical!

Or as you vaguely put it, “seems anti-logical”. It looks like we gotta start with some basics here, beginning with this:

There are no illogical beliefs.

Did you catch that?… That’s right, beliefs are not logical or illogical. Only arguments are logical or illogical, i.e. logically valid or invalid. Logic is about how you get from premises (assumptions) to conclusions, one step at a time in a “chain of inference.”

Logic has nothing whatsoever to do with some particular belief or another.

With that out of the way, I will answer your question about why Kurt Gödel is considered to be one of the greatest logicians ever.

Gödel proved some deep — mind-blowingly deep — results about the limits of formal reasoning systems, results that had catastrophic implications for the foundations of mathematics. What he proved bore directly on some fundamental assumptions of mathematics, and by that I mean things that were basically unconsciously assumed until Gödel showed that they not only can’t be assumed, but actually are demonstrably false! Specifically, it had always been taken for granted that whatever is true in mathematics must be provable. In other words, it went without saying that “the proof is out there… you just have to find it.” Gödel demolished that assumption.

This was in 1931, when he published the proofs of two incompleteness theorems.

The first incompleteness theorem states that if any given computable axiomatic reasoning system is (a) powerful enough to express basic arithmetic (addition and multiplication of natural numbers), or basic set theory (e.g., Zermelo-Fraenkel set theory with the axiom of choice, a.k.a. “ZFC”), and (b) consistent (i.e., unable to prove a contradiction), then there are true statements in the system that cannot be proven by the system. Since we mostly only care about systems that are powerful enough to prove interesting things, and since we are not interested in any system that we know to be inconsistent, this means that the kinds of reasoning systems that we care about are inescapably incomplete. Mathematical truth and provability turn out not to be the same thing after all! (And it gets worse… there are an infinite number of such unprovable but nevertheless true mathematical statements. We just can never know which ones those are, because the only way to know that they’re true would be to prove them… and they can’t be. Proven.)

The second incompleteness theorem, which builds on the first, states that any consistent system is incapable of proving its own consistency.

These two theorems are what Gödel is most known for.

Prior to that, in 1929, he had proved the completeness theorem for first-order predicate calculus. This theorem established that truth and provability are mutually entailed in first-order logic. This is a fundamental theorem of mathematical logic, and has deep connections to model theory. (Actually it’s considered to be one of the foundations of model theory). Gödel did this work for his doctoral thesis at Vienna University.

In the 1930’s, Gödel studied, among other things, the logical relationship of the “axiom of choice” in set theory with a conjecture in the theory of transfinite numbers called the “continuum hypothesis”. So… you know that there are an infinite number of integers, and you also know that there are an infinite number of real numbers, and what you may or may not know is that the infinite number of reals is greater than the infinite number of integers. We say that the integers are “countably infinite”, while the reals are “uncountably infinite”. The number of integers, a.k.a. the cardinality of the set of integers, is an infinite number which we denote [math]\aleph_0[/math]. The number of real numbers is equal to the cardinality of the power set of the integers (i.e. the set of all subsets of integers), so we denote this “cardinality of the continuum” as [math]2^{\aleph_0}[/math]. This infinity is of a higher order or scale than [math]\aleph_0[/math]. Now, the big question is this: is there any infinite number greater than [math]\aleph_0[/math] but less than [math]2^{\aleph_0}[/math]? The continuum hypothesis (“CH”) is the hypothesis that [math]2^{\aleph_0}[/math] really is the “next” infinity after [math]\aleph_0[/math], with nothing in between, and even today nobody knows whether or not it’s true, nor under what set of assumptions it might be true, or even whether it can be said to be “true” or “false” in the usual sense. Anyway, at that time everybody was interested in whether CH could be proved or disproved from the axioms of ZF set theory and whether it made a difference either way if you assume the axiom of choice (“AC”). It is now established that both the AC and CH are independent of ZF, i.e. can neither be proved nor disproved. These proofs were completed by Paul Cohen in 1963, building on Gödel’s earlier work. Gödel had shown that CH — actually, a generalized version of it — cannot be disproved in ZFC, i.e. that it is consistent with both ZF and AC, assuming that ZF itself is consistent. Cohen proved that CH also cannot be proved in ZFC. (Cohen was awarded the Fields medal for that work). Anyway, Gödel’s 1940 paper “Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory” is a landmark is and considered a classic work of mathematical logic.

Gödel was a good friend of Albert Einstein, who helped him to gain his US citizenship. Gödel developed an interest in mathematical physics and formulated the “Gödel metric”, a novel solution to Einstein’s relativity equations.

It’s hard to overstate the impact of Gödel’s work on mathematical logic, the philosophy of logic, the philosphy of science (e.g., via Karl Popper’s epistemology of falsificationism, which is the basis of the concepts of scientific theory and knowledge as we know them today), and indeed, broadly all of Western philosophy since his time, due to his influence on thinkers such as Wittgenstein, the logical positivist Rudolph Carnap (who actually had introduced Gödel to the serious study of logic, in a couple of lectures given in the Vienna Circle, and years later was in turn strongly influenced by Gödel’s work) and many others.

So, yes... Gödel believed in a personal God, and he was also a singularly brilliant logician as well as a world-class thinker in general.

I guess you might be surprised to know that there have actually been a lot of super-smart, logical people who have believed in a personal God. Here are a few more…

• Alonzo Church, another pioneer of mathematical logic. He was one of two people to independently prove the undecidability of David Hilbert’s Entscheidungsproblem. The other one was Alan Turing. If you have ever heard of the “Church-Turing thesis”, “Church numerals”, or the “Church-Rosser theorem”, well he is the “Church” that these things are named after. Many more people have probably heard of the Lambda calculus, which in a sense is the foundation of the LISP programming language and of functional programming languages. Church invented that.
  
  Alonzo Church was a person of deep Christian faith.
  
  Also, there is an academic publication entitled Journal of Symbolic Logic. Alonzo Church was the founding editor.
• Ever hear of a guy named Donald Knuth? If you’ve ever studied computer science, then you have. His PhD, from Caltech, is in Mathematics. His magnum opus, The Art of Computer Programming, is a seminal classic. He’s a living legend and truly one of the pillars of computer science. He may have done as much as any other single person to define the shape of computer science when it was first becoming a discipline, and to set its standards for academic integrity.
  
  Also a committed Christian, who has written about faith and intellect ([2]).
  
  (Incidentally, he’s also well-known as the creator of the TeX typesetting system, which is the forerunner of LaTeX, the system used everywhere for rendering things like mathematical formulae — as above, for “[math]\aleph_0[/math]“, etc.)
• Michael Polanyi, super-smart guy and first-rate deep thinker, not just in his main field of chemistry, but other stuff as well, was also a committed Christian believer. Check out [3] for an interesting read.
• Alvin Plantinga, a Christian philosopher and IMHO one of the most badass logicians of 20th century metaphysics. His mastery of modal logic as he develops the “free will defense” ([4]) has to be read to be appreciated.
• James Clerk Maxwell. You might have heard of this guy… his formulation of what was to become classical electrodynamics laid the groundwork for both Einsteinian relativity and quantum mechanics. He was an intellectual giant by any standard, and he also had a robust Evangelical Christian faith.
• Of course, guys like Descartes, Newton, and Leibnitz. Like the others in this list, they didn’t attain to Gödel’s sheer genius in regard to pure logic. (Though to be fair, these guys were around before the subsequent development of formal logic by Gottlob Frege and others). But they were certainly quite adept at clear, disciplined thinking.
  
  Now, some people will say that these guys (Descartes, etc.) were just sort of “Christian by default”, that everyone was a Christian back then and they just took their beliefs for granted. But that is speaking in ignorance, because the fact of the matter is that these three guys all thought long, hard and deeply about what they believed, and we know this because they wrote about it, prolifically. They were all Christian intellectual activists in a real sense. And not everyone in Europe was in fact a theist in those days. A good deal of Descartes philosophical work was motivated the desire to convince the skeptical naturalists of his own time.
  
  Others will say that Descartes, Newton, Leibnitz etc. can be “forgiven”, so to speak, for their theism because they lived before they could have had the benefits of the knowledge of modern science (cosmology, evolution), etc., and that had those guys known what we now know scientifically, they certainly would have been skeptical naturalists. But that’s fallacious, because none of the areas of scientific knowledge hinted at in these arguments actually has any bearing on the question of whether a personal God exists.

But don’t worry, you don’t need to be super smart or super good at logic in order to believe in God! Plenty of people believe who aren’t either. My point is just that it isn’t hard to find people who believe in a personal God, who are also scary good at logic and know how to handle the sharp tools.

That, and that Kurt Gödel deserves every bit of his rep as one of the all-time badass mofos of logic :-)

Cheers!

[1] Kurt Gödel (Stanford Encyclopedia of Philosophy)

[2] Donald Knuth, Things a Computer Scientist Rarely Talks About

[3] Thomas F. Torrance, “Michael Polanyi and the Christian Faith —A Personal Report” http://polanyisociety.org/TAD%20WEB%20ARCHIVE/TAD27-2/TAD27-2-pg26-33-pdf.pdf

[4] God, Freedom, and Evil: Alvin Plantinga: 9780802817310: Amazon.com: Books

[5] Logic and the nature of God: Stephen T Davis: 9780802833211: Amazon.com: Books (I didn’t reference this in the answer, but I put in in here anyway. Think of it as conditioning to get you ready to read Plantinga :-)

On December 25, 1642, a child was born who would change the world. He went to college were he read natural philosophy which at the time was still based on the work of Aristotle. To supplement his education, he also read the works of Galileo.

In 1687, he published Philosophiæ Naturalis Principia Mathematica, more commonly referred to today as simply Principia. In this book, he lays out the laws of motion, gravitation, and derives Kepler’s law of planetary motion.

This man’s name is synonymous with classical physics and you could not possibly have taken an introductory physics class without hearing his name. Nearly 400 years after his death, his laws are still the first thing taught to all aspiring physicists.

That man was Isaac Newton. For all intents and purposes, he was the first physicist.

It’s pointless to argue which of the two was smarter. They were both geniuses beyond your idea of genius is. Without Newton, there would be no Einstein. We all stand on the shoulders of giants.

Max Planck was a genius too, he won a Nobel Prize, he solved the ultraviolet catastrophe, but as the saying goes “he was no Einstein”. In fact, he won his Nobel Prize before Einstein did. But in 1905, Einstein revolutionized physics with four groundbreaking works. In that one year, he produced four papers that each should have been awarded a Nobel Prize. Ultimately, he received the award for his work on the photoelectric effect. His work on Brownian motion, special relativity, and mass-energy equivalency were fortunately not doomed to be relegated to the dustbin of history.

In the history of physics, there are two names: Newton and Einstein. Everyone else is just everyone else. Second-tier physicists include giants such as Copernicus, Galilei, Faraday, and Maxwell. There are thousands of other geniuses in the field of physics: Planck, Feynman, Alhazen, Navier, Stokes, Fermi, Pauli, Lorenz, Higgs, Noether, Gibbs, Carnot, Thompson, …the list goes on and on, but only two that fundamentally changed the world.

But does any of that really answer the question: why is Albert Einstein studied in schools but other equally smart scientists are not?

Sort of, but not really. In fact, I doubt you study Einstein in school at all. A class on Einstein the person would be fascinating. I wonder if anyone has taught such a course. What I really think you mean is something along the lines of: why do most people recognize the name Einstein and associate it with his famous equation?

The answer to that is actually more difficult, but it’s probably because of his look. Later in life, he had that long, white, unkempt hair that even to this day we associate with the genius, absent-minded professor. Doc in Back to the Future is almost assuredly based on Einstein. Add to that the the camera had been invented, so we have actual photographs of Einstein unlike Newton.

There we go again.

Look, the concept of IQ is misused all the time

An IQ of 160 (or 180 for that matter) doesn’t necessarily make you a genius; nor does an IQ of 125 mean you’re are certainly not a genius.

IQ is an educated guess as to your true potential. It’s quite possible that if two people with an IQ of 160 and 200 respectively meet in adversity or competition of any sort, the 160 IQ person would outwit, outperform and in general surpass the 200 IQ one.

In general IQ Tests are good at separating ordinary people from extraordinary people but not good at separating extraordinary people from one another.

If your IQ is higher than 135 it doesn’t matter; you can, bar some highly atypical case, be whatever you wanna be. If it’s around 125 you probably still can be whatever you wanna be but you’d have to try REALLY hard to succeed in nuclear physics. If it’s below 115 you’re almost definitely not gonna be a nuclear physicist (but you probably wouldn’t want to anyways). If it’s below 85 you might find it kinda hard to follow certain technological advances. If it’s below 65 you are gonna need help. But then again you’re probably not below 65, because you wouldn’t use Quora

That’s all IQ tests can tell for sure. The rest is speculation. You can be a genius with an IQ of 135 or just a very bright person with an IQ of 170. It’s up to you; your motivation; your determination; and how happy and fulfilled you are when pursuing what you want.

Just don’t expect IQ to make you an NBA player if you’re 5′7′’ like me :)

IQ allows you to work through very difficult and complicated problems. Einstein had a fair amount of that, but he had something even more rare: the vision to be able to redefine the whole problem and see a simple elegant way of understanding it.

Take a look at Einstein’s 1905 paper on relativity. The math is trivially easy; any high school student who understands the Pythagorean Theorem can follow it. But the underlying concept, that light always travels at the same speed in the eyes of all observers even if they are moving, seems completely insane at first. We KNOW that’s not how trains and bullets behave. It’s not even clear that you can build a logically consistent model where that’s true, let alone one that matches reality better than Newtonian physics. That was enough to prevent other physicists from even TOYING with the idea. But Einstein could see beyond that. He had some glimmer of intuition that let him say “Well, suppose that were true. What would be the implications?”

And the implications, too, were unbelievable. Moving objects shrink, and get heavier. Time slows down. There is a limit to how fast ANYTHING can move, no matter how much energy you put into accelerating it. Most physicists, reaching these conclusions, would feel that they had descended into madness and should seek psychiatric help immediately. But Einstein could see that they all held together in a consistent framework, and that they could explain many puzzling experimental results.

Few physicists can do something like that ONCE in their lives. But Einstein did it MULTIPLE TIMES:

• He explained Brownian Motion as being due to molecular impacts, and used that to calculate the actual weight of atoms. (At that time, even most CHEMISTS didn’t think atoms were real.)
• He flipped our understanding of light AGAIN (it had already flipped from Newton’s corpuscles to Young’s waves) by showing that light energy appeared to be absorbed and emitted in packets.
• He worked out Bose-Einstein statistics and realized that one photon could stimulate the emission of another photon in phase with it, which is the theoretical basis for lasers.
• He was willing to assume that because acceleration and gravity FELT the same, they actually WERE the same.
• He predicted that there would be a time dilation associated with acceleration and gravity. It was half a century before we could measure it.
• He predicted that the orbit of a body very close to a very large mass (like Mercury near the Sun) would precess. In Newtonian gravity there is no such precession.
• He predicted that matter and energy were THE SAME THING, and hence interconvertible (leading to nuclear energy and weapons). This forced the laws of “conservation of energy” and “conservation of mass” to be modified and merged. (Remember that we had just barely figured out that different forms of energy were the same, and that energy was conserved, a few decades earlier.)
• He pointed out the weird non-local behavior of quantum entanglement in the 1930s, eventually paving the way for quantum computing and quantum optics.

And I could go on. But if you look at each of these (except General Relativity), you will see that the math was fairly easy but the conceptual disruption was massive. GR was probably the one example of Einstein doing very hard, cutting-edge math. He didn’t enjoy it. It was painful. It took him 10 years. But in his prime, Einstein could crank out one completely revolutionary concept per month. Except maybe for Ramanujan in pure math, no other human has come close to that.

Both were tremendously intelligent, but Godel's intelligence was more beautiful. The two could fill a blackboard with equations, but in Einstein's case the distribution was chaotic. Not only were the lines crooked, but he would suddenly abandon a line to finish the equation elsewhere in the blackboard (or even on another blackboard, with the consequent confusion among his students, who were constantly forced to move from classroom to classroom). To make matters worse, inadvertently he would erase with his messy hair what he was writing on the blackboard. Godel's equations, on the other hand, were a paragon of sharpness and neatness. He even used colored chalk, which seemed frivolous to Einstein. Godel's calculations shone for rationality, while Einstein was not willing to be tied down by the laws of arithmetic: he let himself be swayed by the inspiration of the moment. His calculations were intuitive: two and two did not have to make four for him. That's why his equations were so unpredictable and difficult to follow.

Science is bananas!

I honestly don’t understand this question. Particular beliefs have nothing to do with one’s skills in logic, math, or anything else. Same with a particular belief in God or not. There’s nothing logical or illogical about it. Should we consider Darwin any less of a natural biologist who rewrote the book on animal speciation because he also believed in God? Or should we discredit the work of Monseigneur Georges Lemaître because he was a Catholic Priest, which would in fact also include the Big Bang Theory. Oh, and by the way, Aristotle, the guy who first formalized logic, was himself a theist as well. And so was nearly every logician and scientist since until very recently in western academia. If there is anything odd or strange which might need to be explained, it’s why a good half of western academics have abandoned a culturally dominant belief. But either way, I shouldn’t think it would have any detriment to their work. And I certainly don’t think atheism or theism is derivable from either logic or science.

Luck. He was born in right city, right family, right time, and with all the intelligence a fluke of nature could provide. He became smarter due to his education and where life took him, but the raw intelligence was just pure natural luck.

The real question would be how do you make sure humanity is able to find all rare snowflakes like him, and put them in the right path to advance civilization. Sure many like him have been born and withered away due to lack of opportunities of not being born at the right time and place.

Recently I read from Nicolas Kristof, which provides a very clear example.

“I had a visit the other day from Sultana, a young Afghan woman from the Taliban heartland. She had been forced to drop out of elementary school. But her home had internet, so she taught herself English, then algebra and calculus with the help of the Khan Academy, Coursera and EdX websites. Without leaving her house, she moved on to physics and string theory, wrestled with Kant and read The New York Times on the side, and began emailing a distinguished American astrophysicist, Lawrence M. Krauss.

I wrote about Sultana in 2016, and with the help of Professor Krauss and my readers, she is now studying at Arizona State University, taking graduate classes. She’s a reminder of the aphorism that talent is universal, but opportunity is not. The meaning of global progress is that such talent increasingly can flourish.”

It’s a common idea that atheism is more logical and rational than competing belief sets, but that is, as a matter of fact, not the case.

There are things taken for granted in all systems of what people hold true about the world. In logic, they’re called axioms. In strict atheism, “There is no god/God” is one such. In Christianity and other forms of theism, some beliefs will be functionally the same as axioms.

The axioms are what we take for granted. From them can follow belief or disbelief of all kinds. Sometimes they are mutually exclusive, sometimes they are not. Protestants and Catholics differ on sacraments, but hardly on the existence of a higher being, and only rarely on the Trinity. Atheists differ similarly. They are not all skeptics. I’ve encountered atheists that believe in astrology and atheists that are Reiki practitioners and you can probably find an atheist that believes that homeopathic stuff works too.

Axioms cannot be logical. That goes against what the very word means. When we say “that’s logical” we are saying that something like “B follows from A”. But axioms follow from nothing. They are assumed. You are irrational only if your conclusions don’t make sense with respect to your axioms.

It may also be enlightening to look at the history of logic. Let me quote myself, this is an answer to a similar question (in its entirety, so you won’t get more by clicking on the footnote).

At least historically, the assumption of the question, that it is the case that religious people are less logical, is not correct.

That which we call logic was developed by people like Aristotle and Ockham, none of which were against religion. On the contrary. If the religious ones developed the thing, it would seem they feel quite comfy with it.

Without knowing the particulars, I am quite certain it is the same in non-Western cultures.

[1]

Not that we need to go that far back in the annals of logic, and logicians. Alfred North Whitehead was perhaps not religious in any traditional sense, but he certainly was not an atheist either. To my knowledge Bertrand Russell, a highly vocal atheist in his time, had nothing to say against Whitehead’s contribution to their joint work on Principia Mathematica.

CLARIFICATION
Before starting to argue with me about religion versus atheism in the comments, do note that this answer does not take a stand for or against either. If you want to debate atheism/theism, do it with someone who argues about that.

…………………………..

Footnotes

According to Steve Hsu, in a blog post titled “Only he was fully awake”,

My current estimate is that one or two hundred common mutations (affecting only a small subset of the thousands of loci that influence intelligence) are what separate an ordinary person from a vN. There's plenty of additive variance to be exploited, and many desirable human phenotypes that have never been realized.

We should go one step further and ask: what circumstances resulted in von Neumann receiving a genome with so many of these mutations beneficial for intelligence? But here it becomes controversial. While many people acknowledge that the Jews have a long tradition of valuing intelligence and erudition, many fewer dare to speculate that selective pressures within Jewish culture, i.e., smarter Jews being more likely to marry and produce many offspring, may have resulted in a gradual increase in their intelligence over the ages. You can imagine that over many, many generations, these beneficial mutations accumulated in the Jewish gene pool. You can imagine that, eventually, they reached a concentration where it became possible for one lucky person in the early twentieth century to get fifty of these from his mom and another fifty from his dad… an occurrence that wasn’t very likely, but would have been unlikelier still in any other population. And he grew up to become the renowned mathematician, physicist, computer scientist, and polymath we’re now discussing.

I’m trying to make some sense of that question. Einstein is certainly one of the greatest physicists, but that involves a lot of mathematics - practical math that connects with the real world.

Godel was a significant mathematician - abstract math that occasionally connects with the real world.

So your question is something like, is Pavarotti the Babe Ruth of opera?

Comparisons rarely help, except within the same field. Pavarotti vs Jussi Bjoerling….. Even in those cases, where the difference is small, the questions are little more than splitting hairs.

I think we can say that Einstein revolutionized physics. I’d have to ask a mathematician if Godel revolutionized math. But, the Incompleteness Theorem was certainly a game-changer.

But there’s lots of good reasons that Gauss is known as the Prince of Mathematics. (I don’t think anyone got the title “king”.)

I think that without question the answer I would give is David Hilbert.

The aim of Hilbert’s whole program was, I think, to establish a paradox free future for mathematics. There was a foundational crisis in mathematics that had been fully realized by the beginning of the 20th century.

Efforts to found various parts of mathematics in a formal language were leading to paradoxes everywhere.

So my understanding is that Hilbert took, let’s call it for the sake of argument, finitist mathematics, let’s call that sub-area of mathematics F, and proposed that even if it required a larger area of mathematics I, as an instrument to facilitate proofs in F, Hilbert still proposed that it would be possible to prove that the arguments made in I were sound and that I was free of paradox using methods that were purely combinatorial or arithmetical and so acceptable to a finitist. By this reasoning he hoped to show that F could be made paradox free.

More specifically Hilbert proposed to find a specific formal language that was sufficient to express all of mathematics, and that statements in this language would only be manipulated by certain rules. Then he demanded certain things of the system - it should be complete (all true statements should be provable within the system). It should be consistent: no paradoxes should arise, and the consistency proof should use only finitistic methods. There should be conservation, so that any result about “real” objects that was provable using “ideal” objects such as uncountable sets would be provable without using ideal objects. And finally there should be some algorithm which would determine the truth or falsehood of all statements: the system should be decidable.

Then more complicated areas of mathematics such as real analysis could be founded on F, and ultimately all of mathematics could be founded in arithmetic.

Then, Hilbert argued, there would be no need to argue about the actual meaning of the mathematics in the proof system I, as long as it could be shown to be consistent by finitist arguments, or arguments acceptable to finitists. So I could be regarded as a pure instrument.

I think this is the meaning of instrumentalism, in mathematics.

Gödel on the contrary showed that most of Hilbert’s program was simply not realizable, at least for a reasonable understanding of a finitist argument.

So there were two poles in the 20th century at least. Meanwhile mathematics itself exploded in all directions, since some people, being rather unconcerned with all this just went ahead and kept on asking new questions and proving new results and classifying new kinds of spaces and metaspaces. One thinks of Banach’s proof that typical continuous functions are nowhere differentiable.

Instrumentalism in physics is a somewhat similar philosophy, but it has much less grand aims.

Godel was certainly a genius of a high order, and I’d say Wittgenstein was at least brilliant; but Einstein’s accomplishments put him in an even more rarified order.

Here is the paradox that Einstein took on: it seemed that basic laws of physics could simply not be reconciled with electromagnetism, which said that light moved at a certain speed, no matter what the position or velocity of the observer. But for a moving object, v (velocity) + c (velocity of light) logically should add up to MORE than c, the velocity of light. Electromagnetism said “No.” This contradiction greatly puzzled a generation of scientists. No one could solve it. It seemed impossible.

And then Einstein made one of the great leaps that any scientists had ever taken: He hypothesized that the contradictions were totally solvable IF ONE ALLOWED THAT TIME AND SPACE MIGHT WORK DIFFERENTLY FOR DIFFERENT OBSERVERS. But that was a leap far beyond “common sense” or intuition. Because nothing seemed more obvious to human intelligence than the idea that everyone’s clocks, if correctly set, run at precisely the same rate.

My point is, Einstein solved the contradiction at the heart of the physics of his day, by considering something that had been assumed to be wrong for the entire history of the human race. By making these assumptions, freeing up space and time, he made predictions in Special Relativity, and later General Relativity, that proved to be true.

And that may be the bottom line. He came up with theories that:

1. Overthrew centuries of Newton’s Laws, which had become orthodoxy.
2. They were based on something (the idea that time, not light, is flexible) that seemed almost madness.

But then, subsequent experimental evidence demonstrated that his ideas, and not the great Isaac Newton’s, were true!!

And when that happened, Einstein became the darling of the press, and — though he never really sought it — became the symbol of “Genius” in the 20th Century, and has held that position ever since.

The press and the academic world could have anointed anyone as the symbol of Genius…. but Einstein was raised to this distinction because of the great drama he was part of. When the observations were made… one of them depending on measuring the position of a star during an ellipse…. his ideas, in a flash, had triumphed over the other great genius of the ages, Isaac Newton, and ushered in a completely different way of thinking about the basic nature of the universe.

I’m surprised no one picked Leonhard Euler (1707 — 1783)

Note: His last name is pronounced Oiler as in a person who oils machinery.

Leonhard Euler was the most prolific mathematician in history with over 850 publications and much of his work is thought to have been lost (His works collected in 92 volumes are available at The Euler Archive.). He made important contributions to every branch of mathematics then in existence. He founded graph theory and topology and made major and pioneering discoveries in many branches of mathematics including analytic number theory, complex analysis, and infinitesimal calculus. Much modern mathematical terminology and notation comes from Euler (e.g. using f(x) to denote a function of variable x). His mathematical work is so extensive that in order to avoid the confusion resulting from naming everything after Leonhard Euler, his works are generally named after the next person to investigate a particular topic. Euler is also known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory.

I’ll give one result from Leonhard Euler.

Euler’s Formula (there are actually two completely different formulae that go by this name.)

This formula is fundamentally important to complex analysis but that is not why I am introducing it. We can use this equation to get an almost unbelievable result.

Substitute π for θ in Euler’s formula.

Now bring the 1 to the other side and we get

This stunning formula relates the five most fundamental numbers in mathematics in a remarkably simple way.

Von Neumann' is widely considered to be among the top handful of mathematicians of the first half of the twentieth centuries. This puts him in the very rarified air of modern geniuses. Let’s look at a few of his achievements. Few, if any, can match the depth or the breadth of Von Neumann'.

• With Morgenstern, he initiated modern game theory.
• He wrote one of the early books on mathematical quantum mechanics.
• He initiated the original investigations of C* algebras. (Deep and difficult)
• He worked on early uses of nuclear energy, I believe with Admiral Rickover.
• He helped develop the first electronic computer, and the ideas of a stored program.
• He did foundational work on modern set theory, notably the Zermelo-Fraenkel system. Early in career, this was.

He was an expert from theory to practice. There are dozens of anecdotes about him. Here are three.

A. One day at some meeting, a group of engineers approached him with their report of a project they were working. Von Neumann rifled through the pages, and after a few minutes he told them, “It won’t work.” Months later the engineers found him to be correct.

B. In another incident, one morning his wife got a call from Johnny. He was at the railroad station to inquire where he was going? He was, you see, absent-minded.

C. At a party, he got into a disagreement about some arcane points in Byzantine history with a scholar of the subject. Upon a check in the references, it was Von Neumann' who was proved correct.

Yes, the math required for his theories was so complex, he did get help from the top mathematicians to complete his work.

There are a lot misinformed people though, whom, because of the above, insist Einstein was terrible at math. However, Einstein taught himself geometry and was done with it by age twelve, and by age fourteen had mastered both integral and differential calculus.

His math skills might have paled a bit to his theoretical physics skills, but, he certainly was much better than average at math. To put it mildly. And I would bet that if he applied himself to math only, he would have done remarkable things with that, also.

Everyone so far is overlooking Ludwig Wittgenstein, whose Tractatus Logico-Philosophicus (1922) laid the foundations for the Boolean semantics of propositional and predicate logic, motivating Russell & Whitehead, Principia Mathematica (1910, 1912, 1913, 1927).

It's always a stretch to identify any single individual as being "the greatest ___ of all time," regardless what's placed in the blank. If something extremely narrow is put in, it may make sense - like "the greatest one-legged and one-armed tightrope-walking juggler of all time". There have probably been very few - if any - of those, so it would be pretty straightforward to bestow that honor. But broad categories containing large numbers of contenders - logician, athlete, inventor, etc. - are far more difficult, since so many have excelled and contributed in so many different ways.

[Surprised that John Bailey, "avid student of Spinoza, Sartre, and Wittgenstein" didn't cite Ludwig, and pleased to see Robert J. Kolker mention Chrysippus - he's hard to include, however, since so little is known of his actual work in logic .... once Aristotle's approach won the day in logic for the next two millennia, the works of all the other logicians pretty much disappeared, especially that of Chrysippus and the other Stoics, since they were the Aristotelians' main competition.]

Being really good at physics doesn’t necessarily make you smarter than the next guy. Being really good at physics is something that thousands, including me, can claim. I can say from experience that I’m not the smartest guy around.

We tend to grade physicists by their accomplishments rather than their supposed mastery of physics or math. Both Einstein and Feynman did a lot of significant work, but Einstein did more. Each won a Nobel prize, but Feynman’s was shared. Einstein should have gotten two Nobel prizes; he did not get one for relativity, which I find absurd. I’m a huge Feynman fan, but a big part of his notoriety comes from his ability to communicate physics and his quirky personality. As far as I’m concerned, the ranking of physicists starts with Newton at the top, Einstein next, and a host of others, including Feynman, tied for third.

It all depends on your scaling criteria. I think Feynman was the best combination of physicist and communicator of all time, with Gamow getting honorable mention. It’s up to you to decide if that made Feynman the smartest person in the world.

Maybe, but Einstein was a human. John Von Neumann was a Martian. Didn’t you know?

Einstein helped explain the construction of the universe. Von Neumann built a universal constructor. See the difference?

Why do you suppose that Planck isn’t studied in schools? Every high school physics student has heard of Planck.

Of course, Einstein is more famous, a situation that dates back to Eddington’s famous eclipse observation, and that always puzzled Einstein himself.

However, it isn’t really right to say that Planck and other scientists were equally smart. Einstein was the colossus of 20th-century physics. Any one of his major discoveries would have earned a physicist a place in the history books — and he made many of them. General Relativity was so above and beyond that as a professor of mine once put it, without Einstein we wouldn’t have had it for another 100 years.

He was sort of the Beatles of early 20th-century physics, so thoroughly dominant that every other song on the radio was one of his.

To put it in perspective, for Planck to have made a similar contribution, he might have discovered the theory of the photoelectric effect, the uncertainty principle and matrix mechanics, wave mechanics, and then, in 1920, the Standard Model. That’s how improbable Einstein’s accomplishments were.

As brilliant as many of Einstein’s contemporaries were — and many were mind-bogglingly brilliant — Einstein’s was the kind of mind that defines a century.