## Abstract

The central difference between Frege’s and Hilbert’s respective views of mathematics was that on Frege’s view, a mathematical theory must be concerned with a *fixed* subject matter, while Hilbert’s view was that mature mathematical theories presented something like a schema of concepts, and that such a schema can be variously interpreted; as a consequence, mathematical theories are concerned with the common part of *various* subject matters. One of the consequences of this disagreement was that Frege objected to Hilbert’s presentation of the method of proving the independence of propositions in geometry. There is no doubt that Hilbert’s work contributed significantly to a change to a much more abstract view of mathematics. This might suggest that Hilbert’s position dominated Frege’s, and that the latter was left behind with little to contribute. I will counter this suggestion by looking at Frege’s attitude to, and eventual analysis of, the independence results, and his understanding of the formal structure of the Hilbertian axioms. This throws some light on Frege’s attitude to logic as a scientific, but formal discipline.

### Keywords

- Euclidean Geometry
- Completeness Axiom
- Correct Inference
- Continuity Axiom
- Archimedean Axiom

*These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.*

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## Notes

- 1.
The extant correspondence consists of four letters from Frege, and two letters and three postcards from Hilbert. It is published in Frege (1976), with English translations in Frege (1980), Kluge (1971), and also Resnik (1980). In the following, the references will be just to the letters, which can then be easily found in any of the works containing them. In dating the letters, I use Roman numerals to refer to the months, so as to avoid any confusion between European and American date citation conventions.

- 2.
Hilbert’s book is Hilbert (1899b), which went through six further editions in Hilbert’s lifetime, and the seventh (1930) edition has been republished in eight editions since his death. The original edition has recently been republished as Chapter 5 of Hallett and Majer (2004). That volume also publishes for the first time Hilbert’s lecture notes for various courses on the foundations of geometry from 1891, 1893/1894, 1898/1899 and 1902, which are important in understanding the background to Hilbert’s views on geometry, and these views themselves. It should be noted that the lectures from 1898/1899, which preceded the 1899 monograph, are much more expansive philosophically than the monograph itself. Moreover, mimeographed notes of these, prepared by Hilbert’s first doctoral student Hans von Schaper , were widely distributed in the spring of 1899, and it is clear that Frege saw the copy belonging to the Göttingen mathematician Heinrich Liebmann , the son of his colleague Otto Liebmann .

- 3.
For an exposition of some of these results, and of the general approach to geometry behind them, see Hallett (2008).

- 4.
Frege, of course, was concerned with the meaning of the mathematical statements, what it is that allows them to express genuine Thoughts. On the other side, Hilbert stressed the mathematical purposes of the work in his letter of 29.xii.1899 to Frege:

If we wish to understand one another, then we must not forget the quite different nature of the intentions which guide us. I was forced to set up my system of axioms by necessity. I wanted to make it possible to understand those geometrical theorems which I regard as the most important results of geometrical research, …

Hilbert then proceeds to list several results, first among them the classical result of the independence of the Euclidean parallel postulate from the other axioms, and several other new results which require some independence proof or other.

- 5.
See Frege to Hilbert, 27.xii.1899. For further discussion, and further citations from Frege, see Hallett (2010, §1.3).

- 6.
Note that in his earlier letter, Frege had remarked:

I do not overlook the fact that, in order to prove the independence of the axioms from one another, you must place yourself at a higher standpoint, from where Euclidean geometry appears as a particular case of something more general. But the way in which you are proposing to do this appears to me as one which, as things stand, is not workable, for the reasons given. (Frege to Hilbert, 27.xii.1899.)

And in a letter to Liebmann , Frege writes:

I have grounds for thinking that the mutual independence of the axioms of Euclidean geometry cannot be proved. Hilbert proceeds by extending the domain [

*Gebiet*] so that Euclidean geometry appears as a special case. And in this extended domain he can show the freedom from contradiction by means of examples. But only in this extended domain. For, from the freedom from contradiction in a more inclusive domain one cannot conclude freedom from contradiction in the narrower, for precisely in the restriction contradictions may creep in. (Frege to Liebmann , 29.vii.1900.) - 7.
These

*A-*,*B-*geometries are not to be thought of as like the special geometries which Hilbert considers in the course of his work, e.g., non-Euclidean, non-Archimedean, non-Pythagorean, etc.; if Hilbert’s second-level concept is arrived at by involving all the axioms I–V, then Frege’s geometries will*all*be, from Hilbert’s point of view,*Euclidean*, thus, different instantiations of Euclidean geometry. However, for Frege, only one of these is*actually*Euclidean geometry. - 8.
- 9.
- 10.
This reconstruction of theorems as generalised conditionals is foreshadowed in Frege’s letter to Hilbert of 6.i.1900. Frege thanks Hilbert for sending an offprint of Hilbert (1900b), which records a lecture Hilbert held in Munich in 1899, and then writes:

From your Munich lecture, I believe I have recognised your plan still more clearly… . It seems to me that you want to separate geometry completely from intuition of space, and make it a purely logical science like arithmetic. The axioms, which otherwise ought to be guaranteed through intuition of space, and laid at the foundation of the whole structure, are now, if I understand you aright, to be carried as conditions in every theorem, not indeed fully expressed, but rather contained in the words ‘point’, ‘line’ etc. (Frege to Hilbert, 6.i.1900.)

- 11.
That the antecedent is true for Frege is really the reason why he thinks that the consistency of Euclidean geometry in Hilbert’s sense follows from its truth in his sense.

- 12.
The analysis more or less coincides with Carnap ’s treatment of ‘implicit definitions’, which following Schlick , became the term for Hilbert-style definitions via axiom systems. Carnap calls concepts arrived at by this kind of indirect definition ‘improper concepts’. See Carnap (1927). I would like to thank Ansten Klev for pointing out this paper to me.

- 13.
Cf. Boolos (1990, 205).

- 14.
Hilbert’s Axiom I1 says: Any two distinct points always determine a straight line.

- 15.
This was part of much larger project to show that there is some

*spatial*content embedded in DPT. See Hallett (2008, §8.4.1). - 16.
Hilbert’s reference to ‘a-ish [

*azig*]’ etc. is undoubtedly an indirect reference to Frege’s*Grundgesetze*, Frege (1903a, §38). There, as part of a criticism of Cantor’s definition of the real numbers, Frege proposes highlighting the inadequacy of the account by replacing (among others) the terms ‘equal’, ‘greater than’, ‘less than’, ‘sum’, ‘difference’ by ‘azig’, ‘bezig’, ‘zezig’, ‘arung’, ‘berung’ respectively. - 17.
If the structure does not satisfy the geometrical axioms laid down, then the interpretation of ‘point’ will just be some set, and it makes little sense to say that the elements of this set are points.

- 18.
For a general discussion of the ‘analysis of our intuition’, see §8.4 of Hallett (2008), where three case studies are discussed in some detail.

- 19.
The assumption that this

*is*possible had been used by Wallis in 1663 as the basis of a fallacious proof of*PP*. For references and discussion, see Hallett and Majer (2004, 209–210). - 20.
This description would be one of the new definitions of ‘point’ which Frege complains bitterly about: see the letter to Hilbert of 27.xiii.1899.

- 21.
This is perhaps clearest of all in Pasch ’s work. For a brief description, see Hallett (2010, 462–463).

- 22.
This is not the only place where Frege puts forward a formal view of logic, although this is a complex matter, associated partly with his views about how logic and ordinary language fit together. See, among other places, Frege’s first letter to Hilbert, 1.x.1895.

- 23.
For more discussion on the direction Frege’s work might have taken, see Antonelli and May (2000).

- 24.

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## Acknowledgments

I am extremely grateful to Bill Demopoulos for his remarks on this material. Among many other things, I have learnt immensely about Frege from him, both from his writings and from conversation. I am also indebted to discussions with Ansten Klev , especially to his paper (Klev 2011). I would also like to acknowledge the generous support of the Social Sciences and Humanities Research Council of Canada over many years, as well as the FQRSC of Québec, formerly FCAR. As I pointed out in my paper (Hallett 2010), which might be viewed as Part 1 of the present paper, most of the material used here goes back, in one form or another, to 1998.

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Hallett, M. (2012). More on Frege and Hilbert. In: Frappier, M., Brown, D., DiSalle, R. (eds) Analysis and Interpretation in the Exact Sciences. The Western Ontario Series in Philosophy of Science, vol 78. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2582-9_8

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