According to Kant, arithmetic statements such as "7+5=12" are synthetic a priori. Could we alternatively think of this not as a statement, but as an arithmetic-logic operation to be executed (like a speech-act performed by a machine) and whereas the equal sign "=" becomes the result achieved through this act or maybe the final part of the act, as we would type these numbers into a calculator?

If this is plausible, could we place these operations somewhere in Kant's analytic/synthetic - a priori/a posteriori map (e.g. suggesting that synthetic statements are equivalent to performatives, or alternatively suggesting that they are not a priori, but a posteriori because a machine (mechanical, electrical or biological) must finish the act of computation to achieve the result), or would this require a completely different picture which doesn't fit into Kant's distinctions?

  • You don't believe 7 + 5 = 12 unless there's a machine to express it? And the machine may as well be an adding machine from 1950 or a human mathematician since there's really no difference between the two? Am I understanding you correctly? Secondly when I buy groceries and the cash register outputs a paper tape full of numbers, does that represent a "speech act" on the part of the machine?
    – user4894
    May 4, 2017 at 1:10
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    Kant used those terms to refer to judgements as opposed to what you are calling an 'operation' (I'm assuming you mean something similar to the informal notion of computation i.e. 'effective method' etc.) but something you should remember is that Kant said that it is a given that somebody knows what the words of the judgements mean before we decide if they're a priori, etc. Its often phrased as "if a proposition is a priori then someone familiar with the language would know it is true just by thinking hard" so you really have to take the familiarity with meaning into account for your analogy.
    – Not_Here
    May 4, 2017 at 1:40
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    I think that you should develop the definition of 'operation' that you are asserting a lot more and explain exactly how you are relating it to Austin's performativity in your mind (you don't mention it explicitly in your question, just the title). What it seems like you are asking is if the computation of an arithmetical statement can be viewed as either analytic/synthetic or apriori/aposteriori because the computation can be viewed analogously to a speech act?
    – Not_Here
    May 4, 2017 at 1:44
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    See (later) Wittgenstein's phil of math: arithmetical equations are not statements, with content and truth-values; they are rules. May 4, 2017 at 6:08
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    You can see Saul Kripke, Wittgenstein on Rules and Private Language: An Elementary Exposition, Harvard University Press (1982) May 4, 2017 at 8:27


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