The purpose of a mathematical theorem and proof is typically not to demonstrate the truth of some proposition. Rather, it is to demonstrate the conditions under which a proposition does not hold.
For example, consider the notorious Jordan curve theorem, which states that every simple closed curve divides the plane into two disjoint regions: an inside and an outside. The theorem is both obvious and true, but it is notoriously difficult to prove. (Not soooo difficult–we did it as a long homework assignment in complex analysis class in college–but the proof was surprisingly involved, given the obviousness of the proposition.)
So what does the theorem get you? And why is it hard to prove? Because the proposition isn’t always true! Yes, it’s true on the plane. But it’s not true on the torus.
This came up the other day because I was talking with an undergraduate math student about what he could do after graduation. I told him about applied math, and then I was thinking that, after taking a bunch of college math classes, he might be confused about what mathematics research is all about. Theorems are important, but not usually for the reasons that you might think.
I learned this many years ago when reading Imre Lakatos’s classic book, Proofs and Refutations:
The proper order of mathematical reasoning is first the problem, then the theorem, then the proof, with the definition at the very end. The definitions come at the end because they represent the minimal conditions under which the theorem is true. The statement of the theorem itself changes as the proof and definitions develop. And, just as a country is defined by its borders, a theorem is bounded by its counterexamples, which are duals to its definitions. . . .
As an applied statistician, I’m more of a user than a developer of mathematics–in my career, I’ve proven only two results that I’d dignify with the title “theorem,” and one of them turned out to be false. But the same alternation of proof and counterexample arises for me when developing statistical methods and applying them to live problems.
“The definitions come at the end because they represent the minimal conditions under which the theorem is true.”
I had to scratch my head at this a few times, but I think I can make sense of it if you make a distinction between the reasoning of a theorem and the presentation of a theorem. When you’re presenting it, you want to make the definitions and assumptions (and therefore limitations) clear up front; but when you’re still working through the thing, you have yet to settle on what definitions and assumptions are needed, and those will only be settled by the time you complete it, thereby coming at the end.
Or I could still be misreading it.
It can take a long time to figure out the best definitions. If the definition is clever enough, it can greatly simplify proofs. Sometimes we choose definitions for convenience, but often they are the culmination of much effort to figure out the essence of a concept. E.g., definitions of derivative, limit, measure, probability.
I suspect the last paragraph of the post shouldn’t be in italic?
Scratch that, I see it’s on your linked text. Sorry!
I think the evolving nature of definitions is really important, and not just in mathematics. Hilary Putnam used the example of “momentum,” which changed its definition from Galilean to relativistic physics. You can see conservation of momentum as being like the theorem, with the definition being whatever is required to make that true.
In much the same way, terms like “variety” or “fundamental group” changed their definition due to Grothendieck. Even in music, there are complicated questions about how to define simple terms like “chord” (e.g. should note multiplicity count?) In a lot of these cases, you can’t just noodle it out — the most productive definition depends in a complex way on the lay of the larger theoretical landscape.
I believe this is the standard view in fields like the representational theory of measurement, where the goal is to establish necessary and sufficient conditions for certain kinds of quantitative representation.
Imre Lakatos and his ideas/contributions have appeared at least a couple of times on this blog. His mathematics and insight may be formidable but this website puts a different glow on his behavior:
https://www.dropbox.com/scl/fi/7x3yza5rj6y4gue4y02tp/ROH-Lakatos.pdf?rlkey=lufeaiammydclv7gvpr7lqp5c&e=1&dl=0
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I had not noticed this before, but the above contains the name of yet another occasionally appearing character in this blog:
“Yet in a 1971 letter to the psychologist Paul Meehl, Lakatos says that the Communist Party was his last superego and that it was only in 1956 that he finally got rid of that superego.”
I knew that Lakatos was a Stalinist but I didn’t know of the connection to Meehl. I wonder what Lakatos thought of Saul Bellow.
To say that “Lakatos was a Stalinist” is sort of putting it mildly. But, this blog has looked at Lakatos before and this from Phil back in February 12, 2021 is the winning comment:
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Phil on February 12, 2021 8:41 PM at 8:41 pm said:
Decades ago, a Hungarian said this to me: “If you’re walking down the street and run into a Hungarian coming the other way, punch him in the face as hard as you can.” I said “Why?” and the guy said “he’ll know why.”
This is apropos of nothing, really; I’ve just been waiting for years for an opportunity to pass this along and this is the first place that seems like I can squeeze it in.
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This clearly beats the age-old joke regarding the recipe for making a Hungarian Omelet: “Steal two eggs–”
The above witticisms sort of indicate that Hungarians, like many other ethnic groups such as Swedes and Norwegian, have made it in the U.S. and therefore, the criticism is viewed as humorous rather than nasty, Or so I believe. However, I am open to opinions to the contrary.
It is interesting to see the author believes it is a paradox that a “genius” could subscribe to an authoritarian political philosophy, up until an even more powerful authoritarian locked him down in a cage. That made him change his tune I guess.
I’d expect the opposite correlation, if there is much at all. Generally they would think they know better, so adopt a parent-child type view of others.
I forgot to note that, regarding philosophy of science, Lakatos is great.
+1. Elites are more propagandized than the general population, not less.
https://en.m.wikipedia.org/wiki/Manifesto_of_the_Ninety-Three
There are different types of theorems. The Expected Utility theory a la Savage is about what axioms need to be true in order to “measure” beliefs and preferences and yield coherent Bayesian probabilities and preferences. Good luck with that. What about the OG expected value theorem. If certain conditions hold, a DM will prefer A to B iff EV(A)>EV(B). If preferences are transitive, independence axiom holds and continuity, you’ve got a Theorem that is an algorithm predicting behavior.
A implies B does not mean that not A implies not B!
Logically correct. But, often an author will discuss whether the conditions can be weakened.
The Jordan curve theorem is not “obvious”. The problem is that there are non-obvious things that satisfy the definition of “curve”. Another example: The four-color map theorem is actually false unless the conditions are stated carefully.
no. it means not B implies not A. Arthur Conan Doyle. The dog that didn’t bark. Wason’s 4 card problem.
For a less controversial Lakatos historical (and blog-relevant) connection, he prepared for writing Proofs and Refutations by translating George Pólya’s How to Solve It into Hungarian
https://en.wikipedia.org/wiki/Imre_Lakatos
I remember a similar insight when reading Lakatos thirty years ago for a philosophy of science class. There are counterexamples to V – E + F = 2, but not if you define “polyhedron” as a three-dimensional figure that is fully specified by its “vertices”, “edges”, and “faces” (themselves defined in particular ways). You use the proof to exclude figures you can draw but can’t formally define. E.g., a nested cube (a cube with a cube-shaped hollow space inside). The problem here is that the inner cube can be in any position after you decide on the length of the all the edges and fix all the angles where they meet. The figure is simply incompletely defined, so not yet a polyhedron. (There are various ways of fixing this, but it involves adding at least one virtual edge to connect the inner and outer cube, ultimately giving the figure its Euler characteristic of 2.)
Correction: It wasn’t a hollow cube I was thinking of but a figure that looks like a small cube stacked on top of a bigger cube. (Though I think the same point holds for hollowed out solids, maybe for slightly different reasons.)
While I basically agree with Lakatos and you about proofs — I read *Proofs and Refutations* as an maths undergraduate — there’s the range of sharpness of conditions as a limitation. If we have a proof of asymptotic Normality of the ordinary least squares regression estimators based on assuming a constant-variance Normal distribution for Y|X=x we don’t get much of a feel for when the result would hold, because the assumptions are so far away from the boundaries. We’d get a lot more from a more complicated proof under weaker conditions. On the other hand, while we’d like a proof under necessary and sufficient conditions, but even if we had one, it might be too complicated to help us actually understand the boundary between the theorem and counterexamples. A simpler, but less ideal proof might help.
On a positive note, one setting in statistics where I think the proofs help is in understanding when the bootstrap will and won’t work.
Worth pointing out, math has a very strict definition of theorem, as compared to laws in the physical sciences. Consider the two examples of Reimann hypothesis and the theory of gravity.
The RH has been computationally proven true up to the 10^13-th zero. Yet, it’s still considered an open problem. Why? Because it has neither been demonstrated that it’s always true nor there’s a counterexample in which it isn’t true.
Contrast this with the theory of gravity. For large distances, it accurately measures the gravitational force between two large objects. For much smaller objects, it’s considered so weak that it fails. For instance, measuring the gravitational force between two pieces of masking tape, on either finger, gives an absurdly small amount of gravity. The electrostatic force between the two pieces of tape accurately models how they interact, whereas the theory of gravity doesn’t. Yet, the theory of gravity is still considered true.
So, there are other mathematical definitions of truth, not just those defined by theorem and proof.
“The proper order of mathematical reasoning is first the problem, then the theorem, then the proof, with the definition at the very end. ”
I believe that Thomas Kuhn has remarked that, while one might think an investigation begins with definitions, that’s not correct. One is able to create proper definitions only when the investigation has concluded.
Bill:
That last remark is kind of overstating it, though. The definition comes after the theorem, but the effort required to create the definition can lead to insight that changes the problem definition and the proof.
That is, I think of the definition as the logical last step–there, I agree with Kuhn and disagree with the usual presentation of theorems in mathematics textbooks–but I recognize that in the actual development of mathematics there will be iteration between problem statement, theorem, and definition. As is conveyed well in Lakatos’s classic doctoral thesis.