The unknown existence of necessary a posteriori truths in math

In Naming and Necessity, Saul Kripke argues against the view that all necessary truths are a priori. Among the examples he gives to support his argument are necessary a posteriori truths and necessary truths that are not clearly a priori. His first example of the second type is Goldbach’s conjecture in mathematics. While all mathematical truths are commonly held to be necessary, those with unknown truth values are not obviously a priori. I will first discuss the distinctions between necessary/contingent and a priori/a posteriori. After I elaborate on Kripke’s argument, I will provide another mathematical example, the four colour theorem, that appears more convincingly not a priori. However, I will show that it is unknown that either example is truly necessary a posteriori.

First, every truth is either necessary or not necessary (contingent). Necessary truths are truths that have to be the case as long as they have meaning. For example, mathematical truths are thought to be necessary. It is commonly and intuitively held that what is not conceivable is not possible. There is no conceivable situation in which mathematical truths are false without changing what their symbols mean. If the meaning of the symbols changes, then that’s a different truth being referred to altogether. Even in a world without mathematical knowledge by humans or other rational beings, it seems that mathematical truths would still hold because their truth values do not depend on knowledge.

Next, I will discuss the distinction between a priori and a posteriori knowledge. All knowledge is attained in some way, which either does or does not depend ultimately on experience. a priori truths are truths that “can be known independently of any experience,” (Naming and Necessity, 34) which means that they can be proved or derived by thought or reason alone. a posteriori truths are truths that are not a priori, i.e. cannot be known independently of experience. It may be said that the knowledge of any truth depends on experience in terms of the thought and memory processes involved, but what’s important here is where the ultimate justification for the truth potentially lies. For example, mathematical truths are thought to be a priori because even if you use a calculator to calculate some huge result, it still could have been worked out by thought alone. Even if no one does it that way, they can in some finite amount of time.

Kripke points out several complications with the definition of a prioricity that he first gives as quoted above. First, “can be” in the definition is about possibility, which is another tricky concept related to necessity. (34) Second, Kripke questions from whose perspective the experience-independent knowledge is supposed to be possible: God, Martians, or subjects mentally similar to humans? (34-5) To avoid these first two complications, he focuses on current a priori knowledge or beliefs rather than the a prioricity of truths themselves. He decides that “It might be best… instead of using the phrase ‘a priori truth’… to stick to the question of whether a particular person or knower knows something a priori or believes it to be true on the basis of a priori evidence.” (35) Third, Kripke convincingly argues against some philosophers who change “can be” in the definition to “must be”, (35) and I will not address this here. However, I argue that how Kripke addresses the first two complications completely changes the question of whether there are necessary a posteriori truths.

To do this, I shall draw a crucial distinction between ‘known’ and ‘knowable’. Every truth is either knowable or unknowable. a prioricity is a property of some knowable truths. It is not a question of whether truths are known independently of experience as of now, but whether they can be known independently without reference to experience. a prioricity is timeless; if a truth P is a priori, then it is a priori for all time because if P can be known independently of experience now, then it can be known independently of experience under any condition. By definition, saying “P is a priori” is equivalent to saying “P is knowable a priori.” Likewise, “P is a posteriori” is equivalent to “P is not knowable a priori.” However, these are distinct from their ‘known’ counterparts. While “P is currently known a priori” implies that “P is a priori,” “P is currently not known a priori” does not imply that “P is a posteriori.” This is because what we know changes over time, and whether we know that any truth a priori can change over time. What seems a posteriori know may turn out to be a priori when we find a proof or justification that does not depend on experience. Kripke asks only whether a truth is currently known a priori, but that does not resolve whether the truth is a priori, which is an important question regarding the nature of knowledge. Regarding his question of whom the knowledge is relative to, I will consider knowledge in rational minds like those of humans. This is a reasonable restriction considering we have only human knowledge, not necessarily divine or Martian knowledge. In my analysis, this distinction between ‘known’ and ‘knowable’ will come into play.

Let me now relate necessity to a prioricity. Necessity is a metaphysical concept; it regards truths as they are. a prioricity is an epistemic concept; it regards how we know truths. What Kripke argues against is a common position that necessary truths are equivalent to a priori truths, that they are “obvious synonyms.” (38) He does this by bringing up examples of necessary a posteriori truths and contingent a priori truths. I will analyze the former type, in particular mathematical examples, as he writes, “All the cases of the necessary a posteriori advocated in the text [Naming and Necessity] have the special character attributed to mathematical statements… [which] applies… to the cases of identity statements and of essence.” (159)

To motivate the claim that not all necessary truths are a priori, Kripke uses the example of Goldbach’s conjecture, the as-of-now unproved mathematical claim that any “even number greater than 2 must be the sum of two prime numbers.” (36) He takes the conjecture to be necessary by assuming the results of arithmetical computations to be necessary. (36) I take this assumption to be indisputable because results in arithmetic are true by the definition of numbers and rules that relate those numbers, and anyone who tries to reject them would have to change the definitions, which changes the relevant arithmetical propositions altogether.

While it is known a priori that mathematical propositions such as Goldbach’s conjecture cannot be contingent, (159) Kripke suggests that Goldbach’s conjecture itself may be a posteriori. I will elaborate on his reasoning and identify some of his implied assumptions, keeping in mind the distinction between ‘known’ and ‘knowable’:

Kripke says that “we are taking the classical view of mathematics here and assume that in mathematical reality it is either true or false.” (36) Regardless of what he means by ‘mathematical reality’, this assumption should be noncontroversial by the law of noncontradiction, and the fact that the conjecture doesn’t have any paradoxical property like that displayed by the proposition “this proposition is false.” As of now, it is not known whether Goldbach’s conjecture is true or false; there is currently no a priori knowledge of its truth value. If someone believes that it’s true, that’s merely belief, not a priori knowledge.

  1. If Goldbach’s conjecture is true, it is either provable or unprovable. Again, this assumption is valid by the law of noncontradiction.

    1. If Goldbach’s conjecture is provable, for a proof of it to be known a priori, the proof must be either a general proof (with universal claims) or a proof by computation (by checking all even numbers against the sums of all pairs of prime numbers). These are the two types of proofs required for this type of conjecture.

      1. For the first case, Kripke says that whether the conjecture can be proved generally is up to debate, as people have disagreed on whether there is a proof for every mathematical proposition of this sort. (37) There is currently no knowledge about the provability of Goldbach’s conjecture, and so no a priori knowledge about it.
      2. For the second case, recall that Goldbach’s conjecture is a claim about all even numbers. It seems impossible to compute an infinite number of results in a finite amount of time. Kripke says that even though the computation is “of course” possible for an “infinite mind,” he doesn’t know about “finite minds.” (37) It seems impossible for humans given the nature of thought that we are acquainted with. Even if an advanced calculator manages to do it in a finite amount of time, belief in the validity of the calculator would not be a priori because that depends ultimately on experience. (35)
    2. If Goldbach’s conjecture is unprovable, it cannot be known, and so it cannot be known a priori.
  2. If Goldbach’s conjecture is false, it can be proved to be false by identifying an even number without the property. Such a number has not been found, and so it is not known a priori that Goldbach’s conjecture is false.

In each of the above cases, there is no a priori knowledge of the truth value of Goldbach’s conjecture, and so Kripke takes it to be a proposition that is necessary but not clearly a priori. Based on Naming and Necessity, I do not think Kripke would substantially object to my presentation of his argument.

I claim that Goldbach’s conjecture is not a satisfactory example for asserting the existence of necessary a posteriori truths. The argument says that Goldbach’s conjecture is currently not known to be a priori. This raises doubt on whether Goldbach’s conjecture is a priori, but Kripke hasn’t shown it to be not a priori. People who think that all necessary truths are a priori can admit that there are truths whose a prioricity is currently not yet known, but they can still think that the truths would turn out to be a priori if we discovered the ways to prove them. So these people, for whatever reason, would assume that such necessary truths are a priori unless shown otherwise. Kripke’s example does not do this, due to reasoning of the type used for Goldbach’s conjecture itself:

Goldbach’s conjecture is either knowable or unknowable, by the law of noncontradiction.

  1. If Goldbach’s conjecture is knowable, then it is either a priori or a posteriori, by the law of noncontradiction. Either would be a necessary truth since a prioricity is timeless, as I argued above.

    1. If Goldbach’s conjecture is a priori, then there is a proof for it that does not depend on experience. Even if Goldbach’s conjecture gets proved in a way that depends on experience, that’s no indication that there isn’t a proof for it that does not depend on experience. To show that Goldbach’s conjecture is a priori, there must be a proof that there is such a proof for Goldbach’s conjecture, which is potentially more difficult to find than a proof of Goldbach’s conjecture itself.
    2. If Goldbach’s conjecture is a posteriori, then there is no proof for it that does not depend on experience. It must be proved that there is no such proof for Goldbach’s conjecture, which again is potentially more difficult than Goldbach’s conjecture itself.
  2. If Goldbach’s conjecture is unknowable, then it is unprovable. To show this, it must be proved that it is unprovable at all, which again is potentially more difficult than Goldbach’s conjecture itself.

In each of the above cases, it appears harder to know the a prioricity of Goldbach’s conjecture than it is to know the truth value of Goldbach’s conjecture. I think this because Goldbach’s conjecture concerns natural numbers, while its a prioricity concerns knowledge and provability. It seems easier to obtain knowledge about natural numbers than about knowledge and provability. Even if I am wrong about their difficulty, that does not change the fact that neither has been found. This shows that while Kripke can raise concerns about the a prioricity of Goldbach’s conjecture, there are the same concerns about his concern.

Next, I will provide another example that Kripke would probably consider stronger than Goldbach’s conjecture. Note that in my analysis above, Goldbach’s conjecture could be replaced by any claim in mathematics of which the truth value is currently not known. Kripke mentions Fermat’s Last Theorem (36) and the four colour theorem (103) as examples of such claims. It happens that both were eventually proved in the years following Naming and Necessity, so they have gone from conjectures to truths in terms of what we know. The proof of the four colour theorem has a special property that I will focus on.

The four colour theorem was proved with the help of a computer. (Robertson et al.) No proof of it has been found that does not rely on a computer, so as of now the four colour theorem is known a posteriori because the proof depends on the external software and hardware involved. This example seems advantageous for Kripke because the four colour theorem appears to have a truth value while not currently known a priori, and so he might claim that the four colour theorem is necessary a posteriori. However, we must be reminded of the ‘known’ and ‘knowable’ distinction. Just because the current proof of the four colour theorem depends on experience, it doesn’t mean there isn’t a proof for it that does not depend on experience. The same reasoning applies here as it does above for Goldbach’s conjecture. The four colour theorem being known a posteriori right now does not make it a posteriori. Therefore, the four colour theorem is not an example that asserts the existence of necessary a posteriori truths.

If it is shown that there are necessary a posteriori truths in mathematics, that would strongly support necessary a posteriori truths outside of mathematics because mathematics has traditionally been regarded as a field in which all truths are a priori. However, Kripke has not sufficiently shown that there are necessary a posteriori truths in mathematics with his example of Goldbach’s conjecture. Here, I am not claiming that Kripke thinks the example is definitive; I am merely elaborating on why it is not. I have refined the problem to be about the ‘knowable’ rather than the ‘known’ by taking it to be a legitimate problem. The result is that even if conjectures such as the four colour theorem are proved by a posteriori means only, it is not implied that they are a posteriori. So while it may be accepted that mathematical truths are necessary, it remains an open question the a prioricity of mathematical truths that are currently not known a priori.

References

  • Kripke, Saul A. Naming and Necessity. Cambridge, MA: Harvard, 1972.
  • Robertson, Neil, Daniel P. Sanders, Paul Seymour, and Robin Thomas. “The Four colour Theorem.” 8 November 2007. Georgia Institute of Technology. 29 February 2008 <http://www.math.gatech.edu/~thomas/FC/fourcolour.html>.