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An Informal Arithmetical Approach to Computability andComputation, II
Canadian mathematical bulletin, Tome 7 (1964) no. 2, pp. 183-200

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In the first part of this paper [l] there was introduced a hypotheticalcomputing device, the Q-machine. It was derived by abstracting from theprocess of calculating carried out by a man on his fingers, assuming anadequate supply of hands and the ability to grow fingers at will. TheQ-machine was shown to be equal in computing power to a universal Turingmachine. That is, the Q-machine could compute any number regarded ascomputable by any theory of computability developed so far. It may berecalled here that Turing machines were obtained by Turing [2] byabstracting from the process of calculating carried out by a man on someconcrete 'symbol space' (tape, piece of paper, blackboard) by means of fixedbut arbitrary symbols. Hence the contrast between the Q-machine and theTuring machines is that between arithmetical manipulation of counters andlogical manipulation of symbols. In particular, one might say, loosely, thatin a Turing machine, as in arithmetic, numbers are represented by signswhereas in the Q-machine, as on a counting frame, numbers representthemselves. 
Melzak, Z.A. An Informal Arithmetical Approach to Computability andComputation, II. Canadian mathematical bulletin, Tome 7 (1964) no. 2, pp. 183-200. doi: 10.4153/CMB-1964-017-6
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     author = {Melzak, Z.A.},
     title = {An {Informal} {Arithmetical} {Approach} to {Computability} {andComputation,} {II}},
     journal = {Canadian mathematical bulletin},
     pages = {183--200},
     year = {1964},
     volume = {7},
     number = {2},
     doi = {10.4153/CMB-1964-017-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1964-017-6/}
}
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