From this 1981 interview:
So when did the loss of certainty begin? Where did we take a wrong turn?
It began around 1800, and it began with geometry. I usually like to quote Mark Twain about this. He said that man is the only animal that has the one true religion—several of them. And that is just what happened with geometry.
The geometry that came from the Greeks is usually called Euclidean geometry, after Euclid. But suddenly at the beginning of the 19th century other geometries were developed—non-Euclidean geometries. Who gets the credit for this is sometimes disputed among historians, but I would say Carl Friedrich Gauss. He was the man who said flatly that we can no longer be sure that Euclidean geometry describes the physical world correctly. The various geometries conflict, although one of them, according to thousands of years of tradition, should describe the truth. You can see the problem.
Can you give me an example of an alternative geometry?
Well, one can cite as an example the theorem of Euclidean geometry that the sum of the angles of a triangle is one hundred eighty degrees. In one of the non-Euclidean geometries, called hyperbolic geometry, the sum is less than one hundred eighty degrees; in another, called double-elliptic non-Euclidean geometry, the sum is always larger than one hundred eighty degrees. Yet all of these geometries are equally accurate insofar as man can measure the sums of angles of triangles.
I actually disagree with Kline on that point! Draw a (virtual) triangle connecting the North Pole to two points on the Equator that are 90 degrees apart in latitude, and each angle of that triangle will be 90 degrees. 90 + 90 + 90 = 270: we can measure that.
The interview continues:
What did the mathematicians do when the bottom dropped out of geometry so to speak?
Many mathematicians tried to rescue and maintain as truths the portion of mathematics built on arithmetic, which by 1850 was far more extensive and vital for science than the several geometries. Unfortunately, other shattering events were to follow. Arithmetic and algebra were the next to go by the board.
The best example of this I could give in a semi-popular book was the creation of what are called quaternions, in 1843, by the great mathematical physicist William Rowan Hamilton. Now in the algebra of quaternions, a kind of number known as a hyper-number, multiplication is not commutative. In other words, if I were talking quaternions, I could not say that three times four is the same as four times three. Other strange algebras were created, and it made people start to worry about the laws of ordinary arithmetic. (The one I just stated is known as the commutative law of multiplication). And if we can have perfectly good algebras in which the old familiar laws don’t work, then how do we know they work in the case of the real numbers? That’s where a mathematician named Hermann von Helmholtz stepped in and told us we don’t know it at all. They work in some situations, but not in all.
Are there any elementary examples of these sorts of algebras, where 2 + 2 = 6, or where 5 x 7 = 35, but 7 x 5 is only 34?
I can think of several. Take a quart of water at 40 degrees and mix it with another quart of water at 50 degrees. Do you get two quarts at 90 degrees? You do not. It’s more like 45 degrees. So you can’t just say I’m going to add 40 and 50 and automatically get 90. It depends on the physical situation.
Consider music, a simple musical tone with a unique frequency and amplitude, say one hundred cycles per second. Now suppose on top of that you impose another note at two hundred cycles per second. Do you get a note at three hundred cycles? Again you do not. It is a note of two hundred cycles, the first harmonic above the one-hundred-cycle note. It is the highest harmonic that determines the pitch—two hundred cycles. This is an important factor in the design of musical instruments.
Those are excellent examples. And recall that the laws of probability do not apply in real life (that is, quantum mechanics; see section 2 of this article).
This is interesting:
If mathematics has no underlying truth–if it is filled with contradictions and uncertainties, why does it work?
There is no definitive answer to that. It just works. The only test we have that mathematics is reliable–not certain, but reliable–is that one can apply its laws to physical problems and make predictions. If the predictions come through, then we can say that mathematics has some substantial basis, but not certainty. I think people can’t help being impressed by what mathematics achieves. Consider the problem of sending a spaceship to the moon and bringing it back. It is entirely mathematical. Of course, there is a tremendous amount of engineering involved in the production of the ship, but the entire plan for it is mathematical. We have a theory about the sun, the planets, and more distant heavenly bodies. We say that what makes them behave as they do is the force of gravity. But nobody knows whether there is such a thing as gravity. We have no physical understanding of it. The theory is mathematical–gravity is a scientific fiction.
The same could be said about electricity and magnetism, couldn’t it?
That’s exactly right. Everybody today knows what a radio is, and what a TV is, but nobody knows what a radio wave or a TV wave is. You can’t smell one or hear one or taste one. But we do have a wonderful mathematical theory developed in the nineteenth century by the mathematical physicist James Clerk Maxwell. The evidence for this wonderful theory is the performance of our radio and TV sets. So we have to accept the fact that mathematics works, or else abandon our radios and our TV sets.
I pretty much agree but I’d put it slightly differently. There are various mathematical theories that don’t work, and because of that we don’t use them to design radios and TV sets. To put it another way, it’s not quite right to say that “mathematics” works; rather, some branches of mathematics work. For example, you could think of various goofy variants of logic and probability as mathematics, but nobody’s using them to build ships. Or, for another example, physicists don’t use classical (“Boltzmann”) probability in quantum problems, because . . . it doesn’t work. The mathematics that works, that’s the stuff that works. Any bit of mathematics works until it doesn’t, which is the point where people try to push it beyond its bounds of applicability.
I enjoyed this bit:
Are most mathematicians since the loss of certainty now working on these physical problems?
No, they aren’t. Most of the mathematics created today–maybe ninety percent of it–is a waste of time. That is an opinion, but one that authorities who are far more creative and far better known share with me.
Can you give us an example of mathematics you consider a waste of time?Some problems now being considered in the theory of numbers, for example, are a waste of time. Take pairs of primes, called double primes. These are prime numbers in a sequence, eleven and thirteen, for example. No even numbers, of course, are primes. Are there an infinite number of these pairs? Are there triple primes? Endless papers are written about these subjects. Who cares?
That’s how I feel too! That said, I understand that the sorts of insights required to solve this sort of number theory problem are cognitively similar to the sorts of insights required to solve what I would consider to be interesting and important problems in mathematics and statistics. So, even though I agree on “Who cares?”, I don’t think that research in this area is “a waste of time,” any more than it’s a waste of time if you’re an athlete to do cross-training.
The interview continues:
It makes mathematics sound a lot like playing chess or bridge. Exciting, beautiful, challenging; the same words apply to all three kinds of activity.
That’s right. I’m glad you suggested it because it makes the point sharper. People enjoy playing chess. Some people even devote their lives to it. But no matter how ingenious a man is at playing chess or bridge, it isn’t going to change this world one iota. Now mathematicians may probe deeper problems, but it is the same thing.
Again, I kinda feel that Kline is missing the point. For one thing, the effort spent to build programs that can win at chess and go has led to general improvements in machine learning and AI. For better or worse, the study of chess has changed the world, and by more than one iota.
Overall, I’m a big fan of Kline and I like a lot of what’s in that interview, which is one reason it’s interesting to see where we disagree.
“It is the highest harmonic that determines the pitch” — no, it’s the lowest
Agreed. And even if you filter out or very much down the lowest, the fact that you hear all the harmonics of the lowest, makes you identify it as having been generated by the lowest. This occurs often with the acoustic bass. It’s quite hard to reproduce the lowest harmonics, but you hear all the upper ones, and identify it as the low note.
For example, suppose for ease of calculation you pluck a string tuned to 50Hz, you’ll hear 50, 100, 150, 200, 250 and maybe a few more harmonics. But the 50 may be much much quieter than 100, 150, 200 etc. You will still identify this as a 50Hz fundamental pitch (with a particular timbre)
If you plucked a string tuned to 100, you’d hear 100, 200, 300, 400… and would be missing 150, 250, 350 that you’d get from a 50Hz string, and so would associate the note with 100Hz even if the 100Hz fundamental were poorly reproduced.
I think the points on the equator should be 1/4 of a great circle apart, not 1/2 (assuming “opposing” means antipodes). Otherwise you get 360 degrees.
Good catch! I fixed it.
> And recall that the laws of probability do not apply in real life (that is, quantum mechanics; see section 2 of this article).
See also https://arxiv.org/abs/2007.08160
+1 to this, the first section on several methods of drawing from urns is a very clear example of how probability statements about different experiments can’t be mixed.
+1.
Gelman’s specific example is bad, but he is right. Classical probability fails in QM. This is proven by Fine’s theorem. The condition under which the experimental probabilities can be obtained from a single joint distribution is exactly Bell’s inequality, and it is violated experimentally.
See the book “Quantum Foundations, An Empiricist Approach” (https://link.springer.com/book/10.1007/0-306-48047-6)
and also this SEP link and its associated citation to the original paper (https://plato.stanford.edu/entries/bell-theorem/notes.html#note-5).
We’ve explained this so many times in this blog that it is a mystery to me why several people are still confused. Besides the misunderstanding of the physics, there is a fundamental misunderstanding of math: You can’t disprove math theorems by experiment. In math, we prove our theorems.
You can show that certain models or theorems are not relevant with experiments, though. That is what Andrew means.
Maybe it is what he means, but it isn’t what he writes.
Are some people really confused or are they joking?
I’m wondering too. Who’s confused about hyperbolic geometry in 2026?
Fun fact: Bertrand Russell wrote his dissertation about the geometry of the universe. I read it and couldn’t quite follow but apparently he believed that the question of the “true” geometry of the universe could be decided logically and he settled on 3-dimensional Euclidian. A few years before Einstein… it’s funny that this dissertation is hardly mentioned in biographies, e. g. https://plato.stanford.edu/entries/russell.
Of course ever since, Mathematicians have agreed that Mathematical axiomatic systems are an important object of study but can’t in themselves tell us much about empirical reality.
Euclidean geometry is obviously false in some many ways… Take for example Playfair’s axiom stating that exactly one straight line can be drawn parallel to a given line through a point not on that line. Any five years old understands that you can draw lots of parallel lines if the point is big enough.
How do these examples point to a loss in certainty of mathematics? I just not be a very sophisticated thinker. The example of temperature of a mixture in particular comes directly from my field and I don’t understand how it says anything about loss of certainty in algebra.
Apologies,
This was supposed to be a comment to the original article. Not a reply.
Agreed. I think what was lost was the notion that all of mathematics could be deduced from rules that are useful for modeling real world scenarios.
Instead what we discovered is that all of mathematics is rules for generating sentences in some formal language, and there’s a question as to whether the sentences of a given language are relevant for the description of real world phenomenon. And many of the possible coherent languages do NOT have a relationship to physical reality. For example, the language of real numbers and addition does not usefully describe the relationship of temperatures in mixtures of water, whereas it does usefully describe the relationship of mass, or energy.
I *think* this is more or less what model theory is kind of about, but I really dont’ know enough model theory.
Do you mean model theory in mathematical logic? That’s different from what you described.
As I said I dont know enough model theory. But my impression was it was the study of whether a set of sentences in a formal language describes a kind of nonempty idea. That is, does there exist a set of mathematical objects that have the properties described in the formal languages.
it doesn’t have to do with reality necessarily but my impression is if you start from a real world idea, idealize it and construct a formal language to describe it, then you kind of get the model for free, it corresponds to the idealized thing you started with. whereas if you start with idealized languages you are at risk for describing in detail an idea that has no consistency.
its an area I always wanted to read more about but never had a good introduction to.
Model theory is pure math and mathematical logic.
A theory is typically a first-order formal theory. For arithmetic for example, you write down the Peano axioms in the formal theory.
A model is a set where each constant symbol in the formal theory corresponds to an element in the set, and each function symbol in the formal theory corresponds to a function on the set. (I’m trying to be nontechnical, so don’t take this as a precise definition.)
If the axioms are true in the model, then the theorems (derived in the formal theory) will also be true in the model. But, in general, a given set of axioms will have more than one model. Model theory studies stuff like this.
If you take a course in mathematical logic, this will make sense. If you haven’t taken such a course, this explanation is almost certainly unclear. I don’t recommend trying to learn this on one’s own. But, if you really want to, I like “A Mathematical Introduction to Logic” by Herbert B. Enderton. There is a lot of notation and terminology that is unique to mathematical logic, so this isn’t the kind of book where you can start reading in the middle.
The thing is, points aren’t big enough to do that. At least if you’re talking about the same points Playfair was, and not, like, dots drawn on a piece of paper.
When it comes to uses in statistics, math has unbeatable internal validity. Connecting it to some problem involving data, i.e. external validity, is a separate challenge.
I thought that prime numbers and sequences of primes were at the basis of modern cryptography, is that right?
Kline’s whole “loss of certainty” theme is wrong. New theories come with statements about the limits what the theory can and cannot do. Learning about limits is a good thing, not a bad thing. The theories have limits whether you like it or not.
He is like someone complaining that a scientific paper has error bars. Kline would say that the error bars show that certainty was lost. It is better to know something about the uncertainty than to ignore it.
I don’t even think it makes sense to talk about triangles in non-Euclidean geometry without saying what it means to be a triangle and what it means to measure the angles. For example, what Gelman describes as a triangle on a sphere is not the triangle you get from connecting the
edgesvertices in the 3D space in which the sphere is embedded.I’m pretty sure that Kline and Gelman both implicitly moved to the relevant Riemannian geometry on a sphere, so that “straight lines” are great circles and angles are measured in the infininitesimal limit, which means the Euclidean angle between the tangents to the great circles at the point of intersection.
Spoken like a true semanticist
(I agree with you)
Isn’t that just Euclid? A line is the shortest distance between two points; an angle is the inclination of two lines. ? You do have to stay on the sphere, of course, just as in Euclidean geometry you have to stay on the plane if we’re just doing 2 dimensions instead of drawing your triangle out into the 3D embedding space.
The axioms of geometry are the same for Euclidian, hyperbolic and elliptic geometries, with the exception of the parallel axiom, which only holds in Euclidian space and is equivalent to the fact that the angle sum in triangles is 180 degrees.
I don’t want to sound rude, please don’t get me wrong, but I thought people posting here are generally familiar with basic mathematical concepts. This whole thread surprises me a bit.
The very examples in this interview show why weird math does apply to the real world, and the discovery route could have gone from the real world to the weird math.
The best example is non-Euclidean geometry. I suppose it did freak out people in the 19 century, but it’s very easy for my 7th grade math class to grasp. I just tell them that if they are drawing their lines and triangles on a globe, the results aren’t going to be the same as on a flat piece of paper. We don’t go much further than that and flying from Chicago to Petersburg, but they get the idea. And if someone in 1700 had tried to play with geometry on a sphere, he would have come up with some nice results empirically and then maybe Leibniz or someone could have formalized it.
> if someone in 1700 had tried to play with geometry on a sphere
Euler published several works on spherical geometry, does that count? People was definitely not freaked out in the 19th century by the concept of spherical triangles, which goes back almost two thousand years to Menelaus of Alexandria.
Are those examples really excellent though? I think the example Kline gives with temperature is the perfect self-rebuttal: no educated man (with an elementary school degree, that is) would think that mixing a glass of 20% alcohol with a glass of 60% would give 80%, and most certainly this was widely known before 1800 (and long before that too). In the same way the case of “multiple geometries” is really not that frightening. In fact, the only ones who are scared of the uncertainty are the people who can’t learn something new. An actual good example for 2+2=6 would be, of course, Z/2Z; and for non-commutative multiplication the case of O(2) or, if you don’t want to explain what reflections are, SO(3): both of these examples are intuitively obvious to anyone and (yet again!) show that the “uncertainty” is nothing to be feared.
Concerning that “useless theorizing” claim: it is, of course, a widely held and convenient pose. The only problem is that it has always been proven wrong: take “We may as well cut out the group theory. That is a subject that will never be of any use in physics”, or the supposed uselessness of conic sections, or Hertz’s comment on the utility of radio waves, or the fact that number theory is now used in most of cryptography, or any other example chosen from the vast archives of anti-intellectual thought.
It’s saddening to see that bombastic claims and a boorish “practical” outlook are enough for a book to be considered “classic” nowadays. My advice it to use them as your first choice of firestarter whenever at hand. His history of mathematics has a mistake every five pages; his tirade against “New Math” is obvious calumny; and “Loss of Certainty” is just a mix of questionable interpretation, libel on creative mathematicians and appeal to “practicality”. Lastly, let us quote Louis Pasteur: “There does not exist a category of science to which one can give the name applied science. There are sciences and the applications of science, bound together as the fruit of the tree which bears it.”