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Submitted by Richard Zach on Tue, 02/10/2015 - 4:47pm

You may have heard that a notebook by Alan Turing, which he left to Robin Gandy, is going to auction in April. Bonham's, the auction house, has kindly permitted me to share the auction catalog.

The notebook apparently dates from around 1944. The mathematical content is divided into two parts, one on Peano's axioms (judging from the few samples in the catalog, in what looks like *Principia* notation), the other on "notation," specifically variables. He quotes from Weyl's 1939 book *Classical Groups*, where Weyl introduces polynomials as "a formal expression $f(x) = \sum_{i=0}^n \alpha_i x^i$ involving the 'indeterminate' (or variable) $x$ whose coefficients $\alpha_i$ are numbers in a field $k$." Turing writes:

The idea of of an "indeterminate" is distinctly subtle, I would almost say too subtle. It is not (at any rate as van der Waerden sees it) the same as a variable. Polynomials in an indeterminate $x$, $f_1(x)$ and $f_2(x)$, would not be considered identical if $f_1(x) = f_2(x)$ all $x$ in $k$, but the coefficients differed. They are in effect the array of coefficients, with rules for multiplication and addition suggested by their form.

I am inclined to the view that this is too subtle and makes an inconvenient definition. I prefer the indeterminate $k$ [possibly should be $x$?] be just the variable.

Turing's worry is clear enough. If the $x$ in the polynomial is a variable, then the polynomial is determined by its values for all variables, essentially, as a function on $k$. But identity conditions for polynomials are more fine grained. I'm not sure, though, why he thinks this is "too subtle" or why he prefers "the indeterminate be just the variable." Anyway, it *is* subtle, and I don't know if logicians (or algebraists for that matter) by that time commonly made the distinction clearly. When did people start talking about free algebras on generators $\{x, \dots\}$?

The other detail from the manuscript included in the catalog concerns Leibniz's $dy/dx$ notation. Here, too, he seems to be concerned essentially with the role the symbol "$x$" plays in mathematical notation: is it a variable, so that this notation, as he puts it, has "laid down a relation between $x$ and $y$" (i.e., between their values), and that a polynomial is just a special kind of specification of a function? Or are they indeterminate symbols, and polynomials are certain sequences involving these variables? In other words, do the variables belong to the mathematical metalanguage used to describe mathematical objects, or are they symbols in a formal object language?

Frustrating that we don't have more to look at! If you have a few mil lying around and are going to buy this, please share with the poor scholars.

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## Comments

Thanks for this Richard. The Bonhams catalogue shows some very pretty and interesting pictures of Turing. As to your question, I would think that the idea of the free group over a set of generators must go back quite a bit, deep into 19th century mathematics. Related to Turing's worries is the fact that the presentation of the group must be given by "relations" that force the identity between given terms, so that, for instance, a+(b+c) = (a+b)+c in spite of their being obviously distinct terms. These basic identities then induce a quotient over the whole algebra.

Right. But varieties are a 1930s thing, arent they? Birkhoff?

Thank you for sharing this information and for the transcription. I looked at Van der Waerden. There is a significant change in the presenation of the notion of polynomial from the first to the second edition of Moderne Algebra. In the first edition (1930) Van der Waerden defines an indeterminate as a ``Symbol'' and gives as identity criteria for polynomials that corresponding coefficients be equal. In the second edition (1937) he defines a polynomial ring as an algebra over a ring and regards 1, x, x^2, x^3,... as belonging to an infinite cyclic group. He seems to return to the old definition again already in the third edition. In the fifth definition (1960) he at any rate defines an indeterminate as a ``Rechensymbol'' and gives identity criteria as in the first edition. In each edition he also defines the notion of substitution of x with elements from the ring. This gives rise to a function concept and he says (again in each edition) that for this reason polynomials are also called ``ganze rationale Funktionen der Variablen x_1,...x_n.''

Turing's idea of defining a polynomial as an array of coefficients is in effect the definition given of polynomials in Lang's very nice presentation of these matters in his Undergraduate Algebra.

I wonder whether there exists a history of the concept of a polynomial. You find the term `indeterminate' used of x's and y's already in definition of forms in Gauss's Disquisitiones Arithmeticae. He calls such forms `functions' and there seems to have been a tradition in algebra for using `function' for what one today would call a polynomial; so for instance in some of the early papers on Galois theory, and in Dedekind `Funktion' appears to mean just polynomial (he used Abbildung and Operation as terms for what are actual functions).

The notebook sold for slightly over $1M to an unknown buyer: http://www.bbc.com/news/world-us-canada-32295436