There's this thing in the work of Immanuel Kant and Hannah Arendt where they'll slip into Greek and/or Latin, sometimes in the middle of a sentence (even if for just a word there), or sometimes like one sentence in a paragraph will be in one language, the next another, then it switches back, or whatever. If you have some feel for Greek lettering and the meanings of the words, you can track the significance to the other text well enough; in theory, if you could keep your reading flow at the same full rate over all the forms of text, then you'd understand the display of the involved concepts in a full enough way. The mixing of the languages would not alter the "analytical" character of any of the implied answers to "analytical" questions evoked in and by the text.

Now imagine mixing notations for mathematics. Start with Roman numerals but use modern operator symbols. I'll pass over the most basic expressions, here, since I do think they might be analytical, period (it's hard to imagine that 0 + 0 = 0 is not analytical in some impoverished sense, at least). So V × V = XXV, then. Or III × V = XV, or XX × V = C. C × X = M.

Next, replace some of the number terms in Roman numerals with modern numerals: 3 × V = XV, C × 12 = MCC, 9 × X = XC.

So far, the examples have been more streamlined, but so now think of 244 × XIV, or 3972 × CCXII.

Kant already said that his claim about arithmetic being synthetic could be seen more sharply to be true when one considers computations with larger numbers. Although one could, tediously, translate back and forth between the numeral systems to do multiplication of large terms by the row-by-row process from elementary school (even in such a way as that the Roman numerals were maintained in the picture), this might indicate a less-than-analytical character for such computations: analysis should be as direct and "quick" at producing truths as possible, synthesis depends on the vagaries of time and effort.

Especially as we pass to not only larger numbers but more complex sequences of operations, and as we increase the number of languages/alphabets/notations being mixed together in a sentence of arithmetic, do we strengthen our representation of arithmetic as primarily involving synthetic assertions? For many of us, this representation is already strong, but sometimes the simplicity of e.g. Peano or Zermelo format, in identifying the equivalences between additions of numbers, makes arithmetic start to look analytical again.

  • "Analysis should be as direct and "quick" at producing truths as possible, synthesis depends on the vagaries of time and effort"? Computational complexity has nothing to do with "analytic" and "synthetic" as used in the "analytic/synthetic distinction". Synthetic is not synthetic because it is computationally hard, it is synthetic because it requires extra ingredients in addition to the given inputs.
    – Conifold
    Feb 11 at 10:56


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