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I often see mathematicians make statements like

"A priori, it is not clear whether 231283179 is a prime number or not."

This is supposed to mean something like "just by looking at it, without thinking about it, we can't easily see whether 231283179 is prime or not".

Is this a valid use of the phrase "a priori"? It appears to me that most people believe mathematical arguments to be of the a priori kind - and with a fairly simple mathematical argument, one can find that 231283179=191x12109067, and conclude that 231283179 is not a prime number - hence, a priori, 231283179 is not a prime number, and the above quote does not really make sense.

Have I misunderstood, or are mathematicians being lazy?

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    This was already asked, What does a priori mean in a math paper?
    – Conifold
    Commented May 4, 2020 at 20:45
  • This has already been asked.
    – Noah Schweber
    Commented May 4, 2020 at 20:45
  • 2
    And yet no one has bothered to answer it, so we can't close it as a duplicate?
    – hide_in_plain_sight
    Commented May 4, 2020 at 21:35
  • This seems like a botanist asking ‘Is “fruit” abused by cooks?’ The culinary sense of “fruit” is different from, and less precisely defined than, the technical botanical sense; but it’s a perfectly reasonable way of using language.
    – Peter LeFanu Lumsdaine
    Commented May 5, 2020 at 14:16
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    @PeterLeFanuLumsdaine Indeed. I will consider it a helpful abuse of notation, then. Meanwhile, I will do my best to bring prima facie back into fashion - meetings are on Tuesdays.
    – mss
    Commented May 5, 2020 at 15:05

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Your example is just a misuse, even of the mathematical sense of this term -- he means prima facie. But there is a legitimate mathematical sense, different from the philosophical sense.

Various very important philosophers have considered math to be a priori in the philosophical sense. Plato and Kant come to the fore. But most folks have dismissed this concept, either right away, or after looking at the failing of Frege's program, and the resulting need to reframe mathematics. It is hard to believe in mathematical Platonism, and we know various things Kant said about math, especially geometry, are overstatements that cast his other ideas into question.

And outside those (now broken) frames, it seems obvious that any 'a priori' statement about primes involves a silly notion of 'a priori'. Of course, we would not expect our native intuition to have anything to say about some random number and primality. Primality itself is hard to see as an intuitive notion. It is clearly derivative on a long experience with multiplication, and not something that would just pop into the mind of a baby out of nowhere.

At the same time, there is a proper use of a priori in mathematics, that is not quite exactly what Kant would have meant by the term. The notion of continuity seems to be a priori, in this mathematical sense. Babies seem to be able to track faces through space. At a certain age, they notice it is somehow absurd for a face to simply disappear... And the standard of comparison between different actual definitions of continuity has been how well they accord with this a priori notion, which is prior enough to the facility of language that we can't express it reasonably, and we have only really annoying definitions of it that involve infinite smallness, epsilons and deltas, function preimages, the existence of limits, or other obnoxious complexities.

There is a mathematical notion of 'intuition' that is not perfectly like Kant's notion of intuition, but to which the mathematical use of a priori is related in the same way Kant's own definition of the term is related to his own notion of intuition. It reflects the overall notion of 'elegance' or 'simplicity' in math, which are somewhat unrelated to their everyday usages.

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  • ”Your example is just a misuse, even of the mathematical sense of this term -- he means prima facie.” What would you regard as a reasonable use of the mathematical sense? In my experience as a mathematician, the mathematical usage of a priori almost almost lines up with prima facie — “things we can see fairly easily based on current definitions/assumptions (together with previously acquired background knowledge)”, contrasted with “things that take some new non-trivial work to deduce”.
    – Peter LeFanu Lumsdaine
    Commented May 5, 2020 at 14:20
  • @PeterLeFanuLumsdaine What is wrong with the fourth paragraph? What we are 'prior to' in a use of 'a priori' rather than 'prima facie' is often having nailed down an 'intuitive' experience we assume everyone shares in language. No one would say the concept of continuity or succession is 'prima facie'. Otherwise, folks have just gotten their Latin phrases crossed.
    – hide_in_plain_sight
    Commented May 5, 2020 at 15:25
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This is the correct usage by your definition. The instant any further action is taken beyond acknowledging that it is a number one would move out of the "just by looking at it". That number is a plain brown box that may or may not contain "prime". Anything beyond that, is crossing the line. Christmas morning, someone says "this one's for you, but you have to guess what's inside before I hand it over." Your notion that by simply doing the math... is like saying "I'll just open the present a little for a clue", but you can't because it hasn't been handed over yet. You misunderstood your definition. It clearly states that the rules are "no touchy, just looky"

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