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In physical science and math, we encounter some models that can be analytically solved. This means that the properties of models are fully understood and determined by the analytical solutions.

In the literature, we will refer those models as:

exact soluble model

or

exact solvable model

or

exactly soluble model

or

exactly solvable model

Q1. I wonder whether the usage of exact, exactly, soluble and solvable are interchangeable?

Q2. When we say the solutions are analytical, are that both analytic solutions and analytical solutions are correct usage?

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  • Q1: In the technical language of mathematics, 'soluble' is interchangable with 'solvable', the former is British usage, the latter American. In chemistry, 'soluble' means 'can be dissolved in'.

    'exact' is an adjective modifying 'model', 'exactly' is an adverb modifying 'solvable'. So mathematically these are two distinct things (with the assumption that 'exact' has a stipulated mathematical definition. In informal English, these are hardly distinguishable, especially since it is becoming more common nowadays to drop the '-ly' (or put differently, it is becoming more common for the adjective form to be used adverbially).

  • Q2: 'analytical' is more common than 'analytic' in standard English (both meaning roughly the same thing 'able to think clearly and precisely'), but in math 'analytic' is a technical adjective form with a very precise meaning. The only instance I know of for 'analytical' is in the phrase 'analytic continuation'. There may well be other instances, but 'analytical continuation' is all I know of.

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    @ Mitch, I was very interested to see your comment regarding the use of "analytic" in a mathematical context. My experience was that analytical and analytic are also interchanged quite freely in the scientific literature (for example, would you say that "analytical continuation" is incorrect?). Can you comment on the case of dynamical vs. dynamic? – painfulenglish Aug 6 '14 at 7:54
  • @painfulenglish I am only answering for mathematics (I only see 'analytic continuation'). I do agree with you about analytical and analytic in general (that they freely mix). Dynamic vs dynamical is a question to itself. – Mitch Aug 6 '14 at 11:55
  • "Analytical continuation" is actually used quite often in the Google Ngram corpus, and in all the hits I checked it was synonymous with "analtyic continuation". It looks to me from a small sample that "analytical continuation" is much more likely to be in a physics paper than a math paper. – Peter Shor Aug 6 '14 at 12:32
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I was brought up anciently with the notion that problems could be soluble or insoluble, and as an elderly Brit I detest the word solvable, despite its common use in England now. I have to admit that I force myself to use movable and immoveable as active adjectives, when my brain is less injured in enunciating mobile and immobile. Of course, move (moveo) and mobile (mobilitas) are distinguished in modern Latin, though I imagine they have a common ancient root. As an occasional transcriber of shorthand, some time ago I sent a transcription to an American librarian, who complimented me on my skills as a transcriptionist, which somewhat stung may Celtic sensibilities.

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