The discovery of non-Euclidean geometry in the 19th century radically undermined traditional conceptions of the relation between mathematics and the world. Instead of assuming that physical space was the subject matter of geometry, mathematicians elaborated numerous alternative geometries abstractly and formally, distancing themselves from reality and intuition.
The mathematician has only one nightmare: to claim to have proved something that later turns out to be false. There are thousands of theorems in Greek geometry, and every last one of them is correct. That’s what mathematical proofs are supposed to do: eliminate any risk of being wrong. If this armor is compromised, if proofs are fallible, then what’s left of mathematics? It would ruin everything.
But the nightmare came true in the 19th century. What had been thought to have been proofs were exposed as fallacies. Top mathematicians had made mistakes. Mistakes! Like some commoner. It’s going to be hell to pay for this, as you can imagine.
I’m referring to Euclid’s fifth postulate, the parallel postulate. Euclid’s postulate had rubbed a lot of people the wrong way even since antiquity. It sounds more like a theorem.
The earlier postulates were very straightforward: there’s a line between any two points, stuff like that. Very primitive truths. It makes sense that the bedrock axioms of geometry should be the simplest possible things, such as the existence of a line between any two points.
The parallel postulate, by contrast, is not very simple at all. It’s not a primordial intuition like the other postulates. It states that two lines will cross if a rather elaborate condition is met. That’s the kind of thing theorems say. This particular type of configuration has such-and-such a particular property. That’s how theorems go in Euclid. Very different from the other postulates, which are things like: any line segment can be spun around to make a circle.
So, people tried to prove the parallel postulate as a theorem. Many felt that it should not be necessary to assume the parallel postulate as Euclid had done. The parallel postulate should be a consequence of the core notions of geometry, not a separate assumption.
Many people tried to “improve” on Euclid in this way. From antiquity all the way to the 19th century. Around 1800, people like Lagrange and Legendre proposed so-called proofs of the parallel postulate. Those are big-name mathematicians. Their names are engraved in gold on the Eiffel Tower. Lagrange was even buried in the Panthéon in Paris. Elite establishment stuff.
But even these bigwigs were wrong. Their proofs contain hidden mistakes. It’s astonishing that this was more than 2000 years after Euclid. People tried to improve on Euclid for millennia. And not a few claimed to have succeeded. But the fact is that Euclid was right all along. The parallel postulate really does need to be a separate assumptions, just as Euclid had made it. It cannot be proved from the other axioms, as so many mathematicians during those millennia had mistakenly believed.
The Greeks, you know, they were really something else. It’s so easy to make subtle mistakes in the theory of parallels. History shows that there are a hundred ways to make tiny invisible mistakes that fool even the best mathematicians. Top mathematicians who were never wrong about anything else stumbled on this one issue.
Somehow Euclid got it exactly right. He didn’t make any of those hundred mistakes that later mathematicians did. That’s not luck, in my opinion. Arguably, the Greeks were more sophisticated foundational geometers than even the Paris elite in 1800. Unbelievable but true. Euclid’s Elements is not a classic for nothing. Euclid is not a symbol of exact reasoning because of some lazy Eurocentric birth right. Euclid’s Elements really is that good.
When Euclid made the parallel postulate an axiom, he seems to be suggesting that it cannot be proved from the other axioms. And he was right. But, as I said, many people had a hunch that he was wrong about this. They thought it would be impossible for the other axioms to be true and the parallel postulate not true.
So many mathematicians figured they could prove this by contradiction: Suppose the parallel postulate is false. If we could show that that assumption would contradict other geometrical truths, then the assumption must be false. So this way we could prove that the parallel postulate must be true, by showing that it would be incoherent or impossible for it to be false.
Indeed, it was found that negating the parallel postulate had various strange consequences. For example, if the parallel postulate is false then squares do not exist. Suppose you try to make a square. So you have a base segment, and you raise two perpendiculars of equal length from the two endpoints of the segment. Then you connect the two top points of these two perpendiculars. That ought to make a square. In Euclid’s world it does.
But proving that this really makes a square requires the parallel postulate. If the parallel postulate is false, one can instead prove that this construction does not make a square but rather a weirdly disfigured quadrilateral. Because the last side of the “square” doesn’t make right angles with the other sides. So even though you made sure you had right angles at the base of the quadrilateral, and that the perpendicular sides were equal, the fourth and final side still somehow manages to “miss the mark” so to speak. It makes non-right angles.
It’s as if the sides are sort of bent. It’s as if you had four perfectly equal sticks of wood, but then you stored them carelessly and they were exposed to humidity and so on and they were warped. So now they’re kind of mismatched in terms of length and straightness, and when you try to piece them together to make a square they don’t fit right. They make some wobbly not-quite-square shape.
Doing geometry without Euclid’s parallel postulate feels a bit like that. It’s sort of bent out of shape and nothing fits the way it should anymore.
One person who investigated this was Saccheri. He wrote a big book discussing this misshaped square and other things like that, in 1733. Saccheri felt that he had justified Euclid’s parallel postulate by examples such as theses. The square that’s not a square and other such deformities, Saccheri declared to be “repugnant to the nature of the straight line.”
But one might say that he used this emotional language to compensate or cover up a shortcoming in the mathematical argument. He had indeed showed that if the parallel postulate is false then geometry is weird. Then you have squares that don’t fit, and other things that feel like doing carpentry with crooked wood.
But weird is not the same as self-contradictory. Despite their best efforts, mathematicians could not find a clear-cut proof that negating the parallel postulate led to directly contradictory conclusions. This is why Saccheri had to say “repugnant” rather than contradictory. You only get “repugnantly” deformed squares, not direct contradictions such as 2=1 or a part being greater than the whole. Those things would be logical contradictions and you wouldn’t need emotions like repugnance.
In fact, a hundred years after Saccheri, mathematicians came to accept that this strange non-Euclidean world of the warped wood is not contradictory. It is coherent and consistent. It is merely another kind of geometry. An alternative to Euclid.
People used to shout and scream that all kinds of things were repugnant, such as homosexuality, for instance. That doesn’t really prove anything except the narrow-mindedness of those accusers. Mathematicians had been equally narrow-minded. They had tried to justify the status quo for thousands of years. They had tried to prove that their way of doing things–their geometry–was the only right way. Only in the 19th century did they finally realize that it was much more productive to embrace diversity, to accept all the geometries of the rainbow.
For so many years mathematicians could not get away from the idea that the “straight” squares of Euclid were the only “normal” ones, and that the “repugnant” alternative squares of non-Euclidean geometry were birth defects. But they were wrong. Non-Euclidean geometry is as legitimate as any other. It was a creative watershed shift in perspective in mathematics to finally accept this instead of trying to prove the opposite.
Here’s how Gauss, the greatest mathematician at the time, put it in the early 19th century. Negating Euclid’s parallel postulate “leads to a geometry quite different from Euclid’s, logically coherent, and one that I am entirely satisfied with. The theorems [of this non-Euclidean geometry] are paradoxical but not self-contradictory or illogical.” “The necessity of our [Euclidean] geometry cannot be proved. Geometry must stand, not with arithmetic which is pure a priori, but with mechanics.”
Geometry has become like mechanics in the sense that it is empirically testable. The theorems of geometry are not absolute truths but hypotheses like the hypotheses of physics that have to be checked in a lab and perhaps corrected if they don’t agree with measurements.
For example, Euclid proves that the angle sum of a triangle is 180 degrees. But this theorem depends on the parallel postulate, just as Euclid’s proof reveals it to do. In non-Euclidean geometries, angle sums of triangles will be different. So that’s something testable. Measure some triangles to see which geometry is right, just as you drop some weights or whatever in a physics lab to see which law of gravity is right.
Let me quote Lobachevsky, one of the other discoverers of non-Euclidean geometry. Here’s how he makes this point in his book of 1855: “[Non-Euclidean geometry] proves that the assumption that the value of the sum of the three angles of any rectilinear triangle is constant, an assumption which is explicitly or implicitly adopted in ordinary geometry, is not a consequence of our notions of space. Only experience can confirm the truth of this assumption, for instance, by effectively measuring the sum of three angles of a rectilinear triangle. One must give preference to triangles whose edges are very large, since according to [Non-Euclidean geometry], the difference between two right angles and the three angles of a rectilinear triangle increases as the edges increase.” So you need big triangles to tell the difference, just as the earth is round but looks flat from where we’re standing because we only see a small part of it. In the same way we need big triangles to detect the nature of space. Therefore Lobachevsky recommends that we should use astronomical measurements for this: “The distances between the celestial bodies provide us with a means for observing the angles of triangles whose edges are very large.”
Let’s think about the logical structure involved in the realization that non-Euclidean geometry is possible. It used to be thought that Euclid’s parallel postulate was a necessary consequence of the other axioms. Although Euclid seems to have been wise enough to realize that it was not, others erroneously believed that this was a mistake rather than an insight on Euclid’s part.
So the question is: Does the parallel postulate follow from the other axioms? If the answer is yes, then the way to settle matter is to provide a proof, a deduction, starting from the other axioms and ending up with the parallel postulate. So that would be like adding another theorem to Euclid’s Elements.
On the other hand, suppose the answer is no, the parallel postulate does not follow from the other axioms. How then could we prove that? It’s very different in this case. It is no longer about proving a theorem. Rather it is about proving that something cannot be proved. It’s much more “meta” than just proving a particular theorem.
But here’s how you do such a thing. Consider this analogy. Suppose someone believes that all odd numbers are prime numbers. 3 is prime, 5 is prime, 7 is prime, and so on. So someone has become convinced that all odd numbers are prime numbers, and they set out to prove it. The start with what it means to be odd, and from that information they try to prove that that implies that it must be prime as well.
But this is of course wrongheaded. Trying to prove that being prime follows from being odd is just as futile as trying to prove that the parallel postulate follows from the other axioms of Euclid.
How could we set this mathematician straight? How could we prove that what he’s trying to prove is impossible to prove? The way to do this is not by some general proof, but by a specific example.
Look at the number 9. It’s odd, but it’s still not prime. Because it’s 3 times 3, so not a prime number.
The obvious way to interpret this is to say that the guy was wrong with his hypothesis. The claim that being odd implies being prime is false.
But from a logical point of view it is interesting to look at it in slightly different terms. Let’s not think about it in terms of right and wrong. Logic doesn’t care about right and wrong. Logic cares only about what follows from what. When logic looks at a proposition, logic doesn’t ask: is it true or false? Logic asks: does it follow from a particular set of axioms?
Logic is about entailment relations. What follows from what. Logic doesn’t care what assumptions or axioms you use. It only cares about what follows from those axioms.
So in terms of our example with the odd numbers, we shouldn’t focus on the question “are all odd numbers prime numbers?” Instead, from a logical point of view, the better question is: “does being odd entail being prime?” Or “is primeness a logical consequence of oddness?”
We had a counterexample: the number 9. From the logical point of view, we interpret this a bit differently. Not as proving the falsity of the conjecture, because we’re not interested in true or false. Instead, what the example of 9 shows is that it is not possible to derive the property of being prime from the property of being odd.
When we put it this way, we have an answer to that challenging meta question: How can we prove that it’s impossible to prove something? We just did! It’s impossible to prove primeness from oddness. Because if there was a proof that showed that any odd number must be prime, then that proof would apply to 9, since it’s odd, and it would prove that 9 is prime, which it is not. Therefore no such proof could exist.
It was the same in geometry. People thought the parallel postulate was a logical consequence of the other axioms. The way to prove this wrong is to exhibit an example in which the other axioms are true but the parallel postulate is false. Just as in the number theory case we had to find an example where oddness was true but primeness was false.
This is indeed what happened. Mathematicians discovered something that corresponded to the number 9. This proved the logical independence of the parallel postulate, just as the number 9 proves that primeness is not a logical consequence of oddness.
In the geometry case, the role of the number 9 was played by models of hyperbolic geometry. These are visualizations that prove that there are perfectly coherent worlds in which the parallel postulate is false while all the other axioms of Euclid are true.
Once mathematicians started thinking in these kinds of terms, it turned out to be not so difficult to find models like that. Mathematicians really could have done that a lot earlier. Even hundreds of years earlier, or even in Greek times. It’s a bit of an embarrassment that it took so long.
Imagine how embarrassing it would be to sit around for hundreds of years trying to prove that all odd numbers are prime numbers, and ranting about how the very idea of an odd non-prime is “repugnant to the nature of an odd number” only to then discover that, whoops, actually there’s a pretty straightforward counterexample right there, the number 9.
The mistake mathematicians made in geometry was of course not quite so glaring but still in a way it was quite similar. The counterexamples were not that difficult to find. Once mathematicians opened their minds to the possibility of such counterexample, they found them fairly easily.
Mathematicians had missed these rather simple counterexamples for thousands of years because of their closed-minded perspective and preconceived notions. Mathematicians had relied too much on emotions, intuitions, such as repugnance. And they had assumed that there can only be one reasonable geometry, because geometry must correspond to physical space.
Mathematicians could not afford to make those mistakes again. These mistakes are what made the nightmare come true, namely that what mathematicians had thought they had “proved” was actually false.
It was a time for soul searching and repentance. And the lessons from this whole embarrassment were quite obvious. The sources of error were intuitions, such as feelings about how straight lines “should” behave, as well as the notion that geometry means the geometry of the physical space around us.
Those ideas were the losers of the story. The winner was logic. The breakthrough had come by detaching geometry from intuition and reality. By abstracting geometry away to its logical structure only. That was the winning perspective.
To spell out what this means for geometry and its relation to the world, let me quote Einstein’s essay Geometry and Experience. Einstein wrote this is 1921, but he is really just summarizing a standard consensus that had been firmly established decades earlier. But why not use the words of the famous Einstein, they are as good as any to make this point. Here’s what Einstein says:
“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 the properties of real things? In my opinion the answer to this question is, briefly, this: as far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. It seems to me that complete clarity as to this state of things became common property only through that trend in mathematics, which is known by the name of ‘axiomatics’.” That’s Einstein’s word for what I called the logic perspective. Same thing. Einstein continues:
“Let us for a moment consider from this point of view any axiom of geometry, for instance, the following: through two points in space there always passes one and only one straight line. How is this axiom to be interpreted in the older sense and in the more modern sense?”
“The older interpretation [is]: everyone knows what a straight line is, and what a point is. Whether this knowledge springs from an ability of the human mind or from experience, from some cooperation of the two or from some other source, is not for the mathematician to decide. He leaves the question to the philosopher. Being based upon this knowledge, which precedes all mathematics, the axiom stated above is, like all other axioms, self-evident, that is, it is the expression of a part of this a priori knowledge.”
“The more modern interpretation [is]: geometry treats of objects, which are denoted by the words straight line, point, etc. No knowledge or intuition of these objects is assumed but only the validity of the axioms, such as the one stated above, which are to be taken in a purely formal sense, that is, as void of all content of intuition or experience. These axioms are free creations of the human mind. All other propositions of geometry are logical inferences from the axioms (which are to be taken in the nominalistic sense only). In axiomatic geometry the words ‘point’, ‘straight line’, etc., stand only for empty conceptual schemata. That which gives them content is not relevant to mathematics.”
“‘Practical geometry’ [arises if we] add the proposition: solid bodies are related, with respect to their possible dispositions, as are bodies in Euclidean geometry of three dimensions. Then the propositions of Euclid contain affirmations as to the behavior of practically-rigid bodies. All length-measurements in physics constitute practical geometry in this sense, so, too, do geodetic and astronomical length measurements, if one utilizes the empirical law that light is propagated in a straight line, and indeed in a straight line in the sense of practical geometry. I attach special importance to the view of geometry, which I have just set forth, because without it I should have been unable to formulate the theory of relativity. From the latest results of the theory of relativity it is probable that our three-dimensional space is approximately spherical, that is, that the laws of disposition of rigid bodies in it are not given by Euclidean geometry, but approximately by spherical geometry.” That is to say, the angle sums of triangles are more than 180 degrees.
So all of that I quoted from Einstein. But Einstein speaks for basically the entire mathematical community here. He is describing what was, in his time, the standard view that almost everyone took for granted.
Indeed, these points about mathematics turning to pure axiomatics and so on, apply not only to geometry but to mathematics as a whole. Mathematicians took that lesson to heart and never looked back, basically. So the discovery of non-Euclidean geometry was the birth of modernity, you might say, in mathematics. It led mathematicians to conceive their field exclusively in terms of logic and formalism, and forget everything about intuition or the idea that mathematics is linked to physical reality. And that’s pretty much where we are today, almost two centuries later, with few exceptions.
In the 19th century, you could be forgiven for thinking that this was a case of straightforward progress. Mathematicians had simply discovered the right way to do mathematics, or the best way known so far anyway. The new logic perspective was simply better than the old intuitive or empirical stuff. We shedded the old errors like so many superstitions and became enlightened.
Around 1900, that was a pretty credible narrative. The logic perspective had gone from win to win, and done a clean sweep of mathematics. Everything it touched seemed to become instantly clearer and better. Hilbert was a leading mathematician at this time who may be taken as a symbol of this. He turned from field to field and made everything clear and clean and modern with this logical Midas’ touch.
But the winning streak did not last forever. With one knock-out win after another behind him, Hilbert turned to the foundations of the entire subject of mathematics and tried to do the same trick there. Many people were optimistic. The trick had worked every time before, and now the world’s greatest mathematician was going to use it to definitively settle all the questions of the foundations of mathematics, such as proving that mathematics is consistent.
But the trick broke this time, even though it had worked every time before. Hopes of a quick victory proved as delusional as the equally hubristic delusions of the war planners who were marching into the First World War at the same time.
The world came crashing down around the great Hilbert. He was German, and these were not good times for Germany. First the students and younger generation died in the war. Then the many prominent Jewish faculty were driven out of the country. Hilbert’s once vibrant university was quickly turned into a shadow of its former self. Hilbert himself contracted a rare decease for which the only treatment was eating lots and lots of raw liver every day.
1933 was a year of not one but two disasters. The Nazis took power, but there was an equal blow in the world of mathematics, when Gödel proved that the logician’s dream was impossible. Logical formalism could not prove its own consistency. In other words, the program of detaching mathematics from intuition and experience turned out to be inherently limited. Its utopian dream proved to be unreachable, and demonstrably so in fact.
Kant used a beautiful analogy that is relevant here. It goes like this:
“Deceived by the power of reason, we can perceive no limits to the extension of our knowledge. The light dove cleaving in free flight the thin air, whose resistance it feels, might imagine that her movements would be far more free and rapid in airless space.”
Which is of course not true. The dove may think that air causes nothing but resistance, but if all air is removed, the dove would quickly be taught a different lesson of course. Not only would the dove crash to the ground at once, it would also suffocate in seconds.
A similar fate awaited the movement to purge mathematics of intuition and physical content. People like Hilbert were so keen to remove the old dependence on intuition and the physicality of geometry as if these things were nothing but “air resistance” that prevented the flight of pure logic in a perfectly clean vacuum.
But birds cannot fly without air, and neither could mathematics. Gödel’s theorem of 1933 proved that logical formalism cannot prove it own consistency, which in terms of this analogy is like proving that the dove cannot fly in a vacuum.
This setback within mathematics was perhaps just as unnerving to Hilbert and other mathematicians as all those jarring disasters that were piling on in the outside world. It’s cruel joke of history that it had both these worlds collapse at the same time.
Maybe the parallel extends further. World War One was a horror of horrors, but that didn’t prevent us from doing it all over again soon thereafter. And we still don’t know how to get rid of war.
Mathematics has a similar attachment to formalism and logic. As with war, the romantics among us are not too happy about formalistic mathematics. Its power cannot be denied. Some, or maybe even many, of its victories were for the best. But still it does not feel right in one’s heart to drill young people into an army of formalists. Seeing mathematics as nothing but logical inferences from arbitrary axioms is as heartless as realpolitik. It reigns to this day, despite a now checkered record, because the only alternatives are hippie fantasies with no realistic prospects of ruling. Modern mathematics and modern politics are alike in this regard.
Well, that makes for a bleak ending. Perhaps non-Euclidean geometry does not deserve to be associated with all this misery. It’s not non-Euclidean geometry’s fault that mathematicians had made mistakes about the parallel postulate. Nevertheless the impact of the discovery of non-Euclidean geometry on the mathematical psyche was dramatic and long-lasting. It sent mathematicians on a soul-searching bender, the hangover of which is still felt today.