Inuitionist Math & Probability: Riemann Hypothesis Example

The principium tertii exclusi, the principle or law of the excluded middle, what is that? If there is a proposition B then it is either the case that B is true or that B is false. There is no third possibility. Either B = “you, the reader, are an American citizen” or not-B, you are not a citizen. Either B = “you have cancer” or not-B, you do not. Either B = “you are virgin” or not-B, you are not. The possibility of being just a little bit virginal does not exist.

The principle states that this matter of fact is true for all propositions. How do we know the principle holds in all cases? We don’t. We have to accept that it does axiomatically. But what if we rejected the axiom, what then? We enter the realm of intuitionism, a re-thinking of what mathematics is and what it means. According to the Stanford Encyclopedia of Philosophy:

Intuitionism is based on the idea that mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true, and the communication between mathematicians only serves as a means to create the same mental process in different minds.

The Riemann hypothesis, a non-absurd statement about the distribution of prime numbers, is either true or it is not; at least, that is so if we accept the principium. Regardless whether the principium holds, we can say that the hypothesis is not yet proved. What do we mean by that?

Well, starting from a set of simple axioms about the nature of math and some accepted-as-true rules guiding the manipulation of mathematical objects, all deduced paths—strings of statements, where each farther along on the path is true given the ones that came before—none have so far led to the hypothesis. Stated another way, given all the known streams of argument, none have allowed us to deduce the hypothesis.

Given these paths the probability that the Riemann hypothesis is true is neither 0 nor 1. These paths certainly do not say the probability is 0; i.e. that the RH is false. And neither do they say the probability is 1; i..e that the RH is true. If we accept the principium, then we can say that it is true (there is probability equal to 1) that either the RH is true or that it is false. But isn’t a rather strong statement to make, especially considering we must believe it for all propositions?

Let’s remind ourselves that logic is a matter between propositions, and that the propositions themselves are not part of logic. All knowledge is conditional. You can’t say “It is true that B” without adding the condition on which you base this claim. This is a constructivist position. It isn’t true that B = “George wears a hat” unless we construct evidence that makes this so; such as E = “All Martians wear hats and George is a Martian.” Other E are certainly possible. As are E that make B false, or give it probabilities in between 0 and 1.

We can’t say E is true or false or in between unless we offer a different constructive evidentiary proposition relevant to the question. Whether or not such evidence exists is irrelevant to the question whether B is true given E. We accept E is true. Then B is necessarily true given E. Even though, in this case, we have other evidence that suggests E is in fact false.

When we say, for example, that B = “Fermat’s last theorem is true” we imply that there is a condition E which makes it so—even if we do not know what E is. This is important. The civilian saying “B is true” does not know E; he is relying on the premise that his mathematical betters have said E exists. His argument is that experts have said E is true and that given E therefore B is true. The civilian’s argument is therefore either circular or an appeal to authority. But because there is an E that does indeed let us deduce B, this only proves that there are “forms” of fallacies that give true results (this is another argument which David Stove gave).

This is different when we ask what is the probability that the RH is true. Here, we have a jumble of evidence: the beauty of the hypothesis, that many of the consequences of the RH are themselves useful and wide ranging and that other theorems once unproved (but now proved) shared the similar property that its consequences were useful and wide ranging and so on; not all of this is made articulate. Given this evidence we can say that the probability that the RH is true is high. But we’re hard pressed to deduce a quantification for this probability.

What then is the unconditional (intuitionist) probability that the RH is true? There isn’t one. There is no unconditional validity, invalidity, or probability of any argument, not just this one. If our only evidence is that the principium is true, then we begin our argument with a tautology and prefixing any tautology to any argument does not change the validity, invalidity, or probability of its conclusion. If our only evidence is that the principium is false, then we have nothing and can go nowhere. The RH neither follows nor doesn’t follow from knowledge that the principium is true or false. We have constructed nothing so no probability exists.

Finally, the intuitionist turns the question around and asked what the probability the principium itself is true. Given what evidence? is the question we must ask. Since for any proposition B we do not have constructive evidence that either B or not-B, we cannot claim that the principium is always true.


  1. Speed

    Inuitionist Math & Probability … formerly known as Eskimo Math & Probability.

  2. TomVonk

    William you wrote :
    The principle states that this matter of fact is true for all propositions. How do we know the principle holds in all cases? We don’t. We have to accept that it does axiomatically. But what if we rejected the axiom, what then?
    Actually we know a lot.
    Doing mathematics consists to formulate true statements Q. What is a true statement Q?
    There exists a deductive chain A=>B …. =>Q where A are accepted axioms.
    This miraculously works because there is only one case (out of 4 possible) where an implication is wrong, namely “P=True” cannot imply “Q=False”.
    Now let’s imagine that starting with the same A, I construct 2 valid deductive chains where one finishes with Q and the other with non Q.
    Therefore my whole system (not only mathematics, any formal system) contains at least one case where Q and non Q are both true.
    Such a system is called inconsistent and immediately implodes. Why?
    Well because in such a system it is true that for any P and any Q, we have P=>Q.
    In other words anything and everything is true (or false, it is just a matter of convention). Such a system doesn’t allow to derive anything useful e.g is not able to distinguish true and false propositions.
    So you see, the “principium tertii exclusi” is no principle despite the name.
    It is not even an axiom, it is a property.
    It is a property of all consistent formal systems (yes, it looks circular).
    But it stops being circular when one realizes that if one wants to do mathematics, one can only do that within consistent formal systems.
    On the other hand in any other arbitrary system one doesn’t need this property and one can do and say anything in it with the exception of mathematics.
    An example of such a system not having the consistence property is the world of lunatics.
    But you will agree that their system is neither very efficient nor very relevant to interpret the real world 🙂
    A word to the Riemann conjecture.
    Since Gödel we know that no consistent formal system can be complete.
    This theorem shows that things are not as simple as the superficial “A theorem can only be true or false” statement would make believe.
    Actually it means that there are necessarily statements that are true (or false) but that cannot be derived from the axioms.
    And you don’t know who they are and will never know. Never ever.
    So if you make a (any) mathematical statement, it may become in finite time either True (then it becomes a theorem) or False (it becomes falsified) or the Gödel case (you won’t ever know which of both it is).
    The Riemann conjecture could very well be a Gödel case and if it is, it will remain a conjecture forever.
    This case is in a kind of logical “limbo” where the statement eternally floats between true and false like the QM wave function on which one would be unable to ever make a measure.
    It doesn’t help much to know that a measure would give an eigenvalue if the measure is impossible.
    In any case you are right, speaking about “probabilities” in this context is obviously so absurd that the expression “not even wrong” really applies.

  3. Tom Vonk states: “So you see, the ‘principium tertii exclusi’ … is a property of all consistent formal systems … if one wants to do mathematics, one can only do that within consistent formal systems.”

    The Wikipedia seems to disagree: “Many modern logic systems reject the law of excluded middle, replacing it with the concept of negation as failure.”

    My own interest is in the philosophy of the scientific method. Considered as a consistent formal system, where is the principium required? I don’t think it is.

  4. Jan Willem Nienhuys

    If I understand the intuitionist program correctly then the idea is that mathematics is not concerned with true or false. A mathematician’s business is with constructing finite proofs. The logic of intuitionism has been formalised, much to the chagrin of the founder. In this logic there is basically one way to prove “A or B”, namely you either prove A, or you prove B. If “A or B” happens to be given, you can deduce C from it by successively showing that C follows from A and C follows from B, in other words by pretending that you have one of the two possible proofs of “A or B” available, except you don’t know which one.

    In intuitionism you cannot conclude A from “not not A” (the reverse is easy), but the socalled falsum rule applies (from a contradiction follows everything). However, it is a simple exercise to show that not A follows from not not not A.

    A statement that something exists can only be proven (in intuitionism) by exhibiting the something. This has strange consequences. For example, the standard definition of rational number is: x is rational if there exist a nonzero integer N and an integer D such that Nx = D. However it is easy to define numbers that are rational in the classical sense, but not in the intuitionist sense, because there is no known finite construction for N and D. For example D=1 and N=2 if RH is true and N=1 if it isn’t. (And if the RH is proven or refuted, there are infinitely many other possibilities.)

    A consequence is that you cannot prove some x exists with property P(x) by showing that ‘for all x, not P(x)” implies a contradiction (again, reverse is easy). This makes any kind of reasoning with infinite sets problematic.

    Classically you can say that in an infinte set (the integers for instance) there is an element with a certain property or not. Intuitionistically that doesn’t work: you have a finite construction for such an element or a proof that such an element isn’t there. Examining all elements one by one is impossible, so maybe there is a third possibility: there isn’t a finite proof either way.

    So intuitionism says: true and false are unmathematical concepts, we mathematicians only are concerned with proofs.

    Vonk says that (classically) there are statements that are ‘true’ but improvable. How do we know that the Gödel example is a true statement? In the standard proof that follows from the ASSUMPTION that the system (which must contain the integers) is free of contradiction. That the reason why that assumption is guaranteed unprovable in a contradiction free system. Now this type of statement is rather strange, but unfortunately there is no way possible to separate these funny unprovable statements from those that can be proved or refuted. That is Turing’s famous result, proved by using a Turing machine. Supposedly anything that can be calculated at all in a formal system can be done by a Turing machine (Church’ s thesis, not really “proven” but no maner of performing algorithms, including checking mathematical proofs, has been found that contradicts this).

    All this won’t help you with the RH. Anyway, intuititionism isn’t seriously practiced anymore.

  5. Speed

    It seems that all of the above relates to non-deterministic computing.

    In our Renaissance project at IBM, Brussels, and Portland State, we are investigating what we call “anti-lock,” “race-and-repair,” or “end-to-end nondeterministic” computing. As part of this effort, we have built a Smalltalk system that runs on the 64-core Tilera chip, and have experimented with dynamic languages atop this system. When we give up synchronization, we of necessity give up determinism. There seems to be a fundamental tradeoff between determinism and performance, just as there once seemed to be a tradeoff between static checking and performance.

    The obstacle we shall have to overcome, if we are to successfully program manycore systems, is our cherished assumption that we write programs that always get the exactly right answers. This assumption is deeply embedded in how we think about programming. The folks who build web search engines already understand, but for the rest of us, to quote Firesign Theatre: Everything You Know Is Wrong!
    [emphasis addded]

    More at:

  6. Ken

    Aren’t there some quantum particles/physics in which a given “thing” may be different depending on circumstances? E.G., light is an energy wave or a particle (photon).

    Also, the ‘either or’ outlook of the principium tertii exclusi is subject to flawed modeling. Case in point: a person can both be a citizen of the USA and not a USA citizen at the same time…IF…one accepts that having dual-citizenship in some way disqualifies the USA citizenship (as it turns out relative to certain exports, a dual-US & Other national must be considered relative to the non-US citizenship relative to certain types of export eligibility).

    This is noted only to emphasize that labels & definitions, etc. are subject to interpretation — that clear communication of the model, its implicit assumptions, etc. is of utmost significance.

  7. Briggs

    Jan Willem Nienhuys,

    Excellent point re: about ignoring true/false but concentrating on finite proofs. That indeed is the lingo. My claim is that nobody believes this. Everybody instead really does believe that some axioms are true, that some rules are valid, and that proofs lead to true statements. Math is stamped with truth through and through. The concern with intuitionist approaches (my idea is not quite the classic approach to this) is to help decide what really is true and what we can know.

    Even if you don’t believe any of that, it is still the case that we use math/probability to qualify uncertainty, i.e. to say what is true, false, and in between. So even if you passionately reject that a “bunch of squiggly symbols” has anyhing to do with truth, other things do. For example, our conversation. We are arguing now about what is true, what isn’t. If you convince me that my modified version of intuitionism is false, then we have said a certain thing is false, and we have not just constructed a finite proof. Or if I convince you it is true, etc.

    The largest point I wish to make is the conditionality of all logical/mathematical/probabilistic statements. No proposition is true/false/probable unconditionally; we have to construct evidence which makes the proposition true/false/probable.

    If I can convince you of the truth of that, then we’re getting somewhere.


    Here is a test of $\latex$, which is now supposed to be enabled. Just use single dollar signs to encapsulate any mathematical equation, just as normal.

    $\int e^x dx = e^x + c$

    [latex]\int e^x dx = e^x + c[/latex]

    Hmmm. Isn’t going. I’ll keep working on it. When you see this turn into a real equation, you’ll know it’s up.

  8. Rob

    It’s interesting that you bring up the Riemann Hypothesis in the context of probability. There is an argument by Denjoy based on the Mobius function “looking like” a series of random coin tosses that the Riemann Hypothesis is “true with a probability of 1.”

    See this article and look for “Denjoy’s probabilistic argument.”

  9. Briggs


    Not only that, when we say something like “prime numbers are random” we mean, given the evidence we have, we cannot predict where they/the next one will turn up.

  10. Briggs


    In a way, axioms themselves are statements known to be true but not provable. In that context (and in retrospect), Godel’s results are not unexpected.

  11. TomVonk

    I have mentionned Gödel only anecdotically referring to the RH to show that there is no warranty that this mathematical statement will be ever proven true or wrong .
    But there is of course a big difference between an axiom being postulated true and the much deeper Gödel theorems. The fact that it was an attempt to answer a Hilbert problem which was considered as the most important unsolved problem in mathematics shows that the result was neither intuitive nor expected (even in hindsight).
    However my main point was the answer on your question “Why principium tertii exclusi?”
    My answer is : because it is necessary.
    Necessary for what? For doing mathematics as we have developped it during the last some 3 000 years (Peano arithmetics etc).
    Of course my answer doesn’t extend to all arbitrary formal systems – it is valid only for mathematics.
    There is an infinity of formal systems – you just choose a set of symbols and a set of rules how to combine them.
    These rules are called logics so there is obviously an infinity of possible logics.
    And also an infinity of inconsistent formal systems (e.g those which don’t have the principium tertii exclusi property).
    So in some very general and sterile way, one can define an infinity of formal systems where “anything goes” and in which there is no sharp difference between the “true” and “false” property of statements.
    I would even bet that among this infinity of sterility some amusing intellectual games may appear.
    But the point is that none of these (inconsistent) systems can reproduce mathematics as we know it.
    Symetrically, and I don’t know if it has been proven but it “feels” right, I would say that most (all?) consistent formal system (e.g those which have the principium tertii exclusi property) is isomorphe to the mathematics as we know them.
    In that case mathematics would be equivalent to a class of consistent formal systems differing only by an isomorphism.
    So as long as you want to do mathematics, you have no choice, you need consistent systems.

  12. Outlier

    I have just one little quibble about this. We are talking mathematics with principium tertii exclusi, but the first paragraph dilutes this with popular examples with non-mathematical contexts. “American citizen” is defined by the US laws, which change over time and were definitely not written by a mathematician. “you have cancer” T/F is for practical purposes an irrelevant question since it cannot be answered “false”. “you are virgin” is a very serious question in some cultures, so beyond the usual vagueness of the meaning of this question, there is quite a body of work about how poor the usual tests for this are. The other popular saying “you are either pregnant or not” is popularly quantified by asking “how many months?” so there are degrees or shades to this question after all. In the real world there is always a context, so there is always a condition.

  13. Briggs

    My dear Outlier that there is always a condition is the point (math or otherwise).

  14. Outlier

    Yes, my quibble was with the examples, not with the logic.

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