# Why are the standards of mathematical proof still unsettled?

Please see the embold phrases below. I'm just a laywoman, and I'm just seeking simple answers. I last took math when I was 17.

1. I read Has the standard of mathematical proofs changed over time?, but how did the standards of mathematical proof change "in the millennia separating Euclid from Hilbert"? I don't understand how they can change, when logic is forevermore permanent and objective.

2. Why are "standards of mathematical proof" "still not quite settled"?

### RATIONALITY IN FLUX

Economic analyses have been based on Plato’s notion of a skillful weigher: To evaluate a plan, one first assigns utilities to every type of pain or pleasure that may result from the plan, at every point in time; then weights the utilities by probabilities to take account of uncertainty; and, finally, evaluates the plan in terms of such a weighted sum. My arguments indicate the need for a broader standard of rationality. Why should the tandard not change? Even standards of mathematical proof changed in the millennia separating Euclid from Hilbert, and they are still not quite settled. [Emphasis mine] I see no reason why Plato’s standard of rationality—or its modern economic versions, for that matter—should not also change in the face of increased knowledge of decision making, especially knowledge about choice construction and context-dependent social goals.

Paul Slovic, The Irrational Economist (2010), p 70.

• An example: calculus has been used since the 1600s, but rigorous definitions of some of its basic concepts (such as limits) were not formulated until the 1800s. The lack of careful definitions in certain areas of math led to confusion between distinct concepts that were not recognized as distinct (an example is pointwise and uniform continuity) and to proofs of results that are now regarded as false. Some subtle issues of mathematical logic (related to the axiom of choice) were only discovered in the late 1800s.
– KCd
Jun 6 at 5:06
• Have you thought about computer generated proofs? In 1976 Haken and Appel published their proof of the Four Color theorem and in the following decades things have become much more complicated, sophisticated and contentious. Jun 6 at 9:49
• logic is forevermore permanent and objective --- Useful googling terms: temporal logic AND relevance logic AND modal logic AND intuitionistic logic AND paraconsistent logic AND provability logic AND (OK, enough said!) Jun 6 at 15:54
• 1. Euclid made assertions that are (in modern eyes) not justified. If point $A$ is inside a circle, and point $B$ is outside, then there is a point of the line segment $\overline{AB}$ that is exactly on the circle. Hilbert formulated foundations for Euclidean geometry that include justification for such things. 2. I think you will have to ask Paul Slovic what he means. Jun 6 at 16:15

First of all, Slovic is a psychologist, not a mathematician. Most likely, his knowledge of modern mathematics is quite limited. (Modern here means, roughly, developed in the last 100 years.) I would be very surprived if he were able to read any papers published, say, by Annals of Mathematics (which is widely regarded as the best math journal) in the last 80 years. Hence, he should not be regarded as an authority in math matters.

Standards of mathematical proof (at least in pure math) remained essentially the same for the last 100 years. (David Hilbert was probably the one most responsible for the current standards.) There are several important exceptions here:

1. Computer-aided proofs mentioned by sand1 in a comment. The first computer-aided proof I know, appeared in the solution of the 4-color problem (mid-1970s). Another, more recent, example is the solution (in mid-1990s) of Kepler's problem on optimal ball packing in the 3-dimensional Euclidean space. There is no commonly accepted protocol on when to regard such proofs as solid. (The issue is that it's hard to check correctness of long computer codes.) As for why, the answer is relatively clear: Technological developments outpace our capacity to formulate the standards.

2. Some proofs are incredibly long, complex and verifications require specialized knowledge that very few mathematicians have. When to regard such proofs as solid is still unsettled. (Relatively) recent work of Shinichi Mochizuki on Inter-universal Teichmüller theory is one such example. (I am not going to take sides in this controversy, this is too far from my area of research.) As to why, a partial answer is the increase in specialization, when even best experts working in the same area of math do not completely understand each other. Another part of the answer is that it is only recently that we started to see proofs which (when written in detail) are over 500 page-long.

However, keep in mind that (so far) these exceptions apply only to a very small fraction of mathematical results. Hence, overall our standards of rigor in math are currently settled.

Edit. I looked up David H. Krantz. Yes, he indeed was a math major at Yale. He published just one paper in pure math (after getting his BA from Yale) and after that, all his math-related papers are in applied math (I counted 19, in applied mathematics), where standards of rigor are quite different from pure math. My remarks regarding Slovic, apply to him as well.

• Slovic didn't write this chapter. I forgot to write this, but this quotation hails from Ch 7, Constructed Preference and the Quest for Rationality, by David H. Krantz, who also isn`t a mathematician. But "David H. Krantz graduated from Yale University (Mathematics) and received his Ph.D. from the University of Pennsylvania (1964, Psychology)." Jun 11 at 5:19

The statement you cite is somewhat misleading. If you compare Euclid or Archimedes with modern mathematical publications, there is almost no difference in the standard of rigor.

When they say that "standard of rigor changes", they usually mean the epoch from 17 to 19 centuries when calculus was invented, and its practitioners were impatient to obtain new results, instead of working on foundations. But even at that time, best practitioners of calculus understood, that the rigor of their proofs was inferior to that of Archimedes.

Same happens in modern times. When a big discovery is made, people sometimes care less about rigor and hurry to obtain new results. If these results are important, other people work on making the proof rigorous. For example, it took about 200 years to develop rigorous foundations of calculus (which would satisfy Archimedes).

By the way, when we are speaking of Archimedes, he also obtained non-rigorous results, based on physical intuition. But he had a clear understanding of this, and carefully separated his rigorous results from non-rigorous ones.

I do not completely agree with Alexander Eremenko's answer (though I understand his point - just hope the following does not sound too provocative).

Was Italian algebraic geometry at the beginning of the XXth century providing proofs of their statement? Yes, certainly, though a number of them turned out to be wrong, therefore not rigorous. And yet their arguments revealed a good amount of informations on "true" algebraic varieties once some additional hypothesis were added. Certainly at the time there were published there was wide consensus on the fact that those proofs were rigorous.

Were papers published in many URSS journals between the 50ies and 80'ies rigorous? Proofs usually were given in the form of very brief hints on the line of thought of the proof and details were abundantly left to the reader. Still there was a widespread consensus on the fact that those can be considered proofs, sketchy as they were (and some of them turned out to be not completely correct after all). Would they have been accepted in a Western math journal without questioning their correctness at the time? Probably not, so that one should say that the standard of rigor was at least country-dependent.

A proof is a couple of different things. A completely clear and rigorous set of steps from some hypothesis (and under some agreed upon set of rules which, as Kopylov remarked, were never completely agreed upon, see the dispute around Axiom of Choice). But in this sense proofs exists only in dreams - or rather, at present, in computer-verified proofs. Then, also, they are a way to communicate to other mathematicians a line of thought, more or less detailed, around which a community agrees that it can be turned into a rigorous and complete proof. And here you see my choice of words: "communicate", "community", "agreement" - it is partly a social issue and as such it changes with time and place.

Is a "proof" what it is supposed to be if there is no one on Earth able to comprehend it? Not so sure about it. Mochizuki's proof of the "ABC" conjecture comes to mind here. But what about the classification of finite simple groups? Are we sure it is proved if after its "proof" so many different mistakes were found and fixed, or do we simply just believe in it?

Is the authority of who states the proof really, in practice, not relevant? Are we really ready to accept that an argument is a proof if it comes from an unknown amateur researcher with no affiliation and no previous research record than we are ready to accept it from a Fields medalist? And how it comes so, if proofs are carved in marble for eternity?

How it comes that we feel safer about a statement in math if we know that it can be proved with different techniques? A statement is true or false, after all: but in our research experience we remember way too well of that many times that we strongly believed that some math statement was true (as a consequence of very convincing arguments) and later realized it was wrong.

So my answer to this question is: a math proof is always, partly, a "social contract" and from this perspective no doubts that its standards changed and will keep on changing.

The reason is that there is no analogue of Church–Turing thesis in logic.

Let me explain. In computational theory there is well-known Church–Turing thesis that states that all reasonable definitions of computation are equivalent to each other. That is, all possible definitions of computation (like Turing machine, lambda-calculus, Minsky machine) turn out to be equivalent: they define exсctly the same set of computable functions. So there is one natural definition of computation, you don't need to pay attention to details.

This is somehow misterius. But what is much more misterius, is that there is no such thing in logic! There is no one way to define what proofs are and what is provable. Every historical attempt definitions of proofs turned out to define different set of theorems. There are constructive proofs and non constructive proofs (and there are many flavors of constructivism). But this is not all. You can use set theory or type theory or some more exoctic way to define mathematics. They are not equivalent! Suppose you decide to pick one - say set theory. But there are many set theories like ZF, ZFC and so on. Even if you settle on your favorite theory, like ZFC, then there still many variants in details of definion that may lead to different set of theorems.

So there in no any natural universan definition of mathematical proof.