Abstract
In the late nineteenth century, Pasch made a well known statement concerning the conditions of attaining rigor in geometrical proof. The criterion he offered called not only for the elimination of appeals to geometrical figures, but of appeals to meanings of geometrical terms more generally. Not long after Pasch, Hilbert (and others) proposed an alternative standard of rigor. My aim in this paper is to clarify the relationship between Pasch’s and Hilbert’s standards of rigor. There are, I believe, fundamental differences between them.
To proceed axiomatically means nothing other than to think with awareness (mit Bewußtsein denken)
Hilbert (1922), 201
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- 1.
Roughly speaking, an expression E may be said to occur in a class of expressions \(\mathcal {K}\) if (i) E is an element of \(\mathcal {K}\) or (ii) E is an expression upon whose meaning the meaning of an element of \(\mathcal {K}\) depends.
- 2.
There have also been important alternative conceptions of rigor concerning which failure of rigor is not conceived as it is conceived here. One such conception is what I have elsewhere referred to as probative rigor. This is rigor which, roughly speaking, concerns the extent to which everything adverted to in a proof that is in some sense capable of being proved is in fact proved. Both Bolzano and Dedekind, as I read them, advocated probative conceptions of rigor, although the particulars of their conceptions were different. For more on this and related matters, see Detlefsen (2010, 2011).
- 3.
There are distinctions between different types of inferential failures that should be borne in mind here. Among these is failure to recognize that premisory importation has occurred when it has occurred. Related to, though also distinct from, this type of failure is failure properly and correctly to identify the premise(s) imported. These seem to be distinct types of failures though their differences will not feature in what follows.
- 4.
Conceived as I conceive them, inferring agents include not only those who may devise a given piece of reasoning, but those who, though they may not devise it, nonetheless judge it to be valid.
- 5.
Proper, that is, for purposes of judging the validity of \(Inf_{Id}\).
- 6.
What is fundamentally wrong, then, with judging a premisorily surreptious argument to be valid is not that, taken to include its surreptious premises, it is not valid. Rather, it is that premises sufficient to warrant a judgement of validity have not been properly identified or registered as premises.
- 7.
- 8.
There are at least two different ways to understand gapless retention of subject. One is to emphasize a notion of awareness, and to take gapless retention of subject to consist in some type of continuity of the objects of awareness of a prover throughout the course of a proof.
Gapless retention of subject might also be conceived along more logical lines. On such a view, proofs would be seen as characteristically having parts—in particular, constituent judgments and inferences. Each of these parts would itself have a subject, and gapless retention of subject throughout the course of a proof would consist in the subjects of the relevant parts of a proof standing in a certain relationship to each other (e.g. being identical to or in a relevant sense continuous with each other) and to the overall subject of the proof.
Of these two broad understandings of gapless retention of subject, the latter might seem the more appealing. On the surface, at least, it would appear to allow that gapless retention of subject be an objective matter. This may be deceiving, though, in that it is possible that any satisfactory understanding of the central notion of a proof’s having a subject would have to make use of a subjective element, perhaps in the form of an appeal to a prover’s awareness. There may ultimately be no other way to make sense of the idea of a proof’s being about something that a prover, in order properly to be a prover, must associate with it as its subject.
- 9.
Despite what this passage may suggest, Poncelet’s endorsement of traditional synthetic procedure was qualified. He seems particularly to have had reservations concerning its laboriousness, which he saw as being primarily due to a perceived need for the prover to take things back to rudimentary constructions—or, as he put it, “to reproduce the entire series of primitive arguments from the moment where a line and a point have passed from the right to the left of one another, etc.” (ibid.).
- 10.
Presentist standards of rigor seem to have been familiar to writers well before Poncelet’s time. My reason for mentioning him is to indicate the influence that such ideas still had on nineteenth century mathematicians.
An older description of presentism, and (some of) its supposed virtues, can be found in Berkeley.
It hath been an old remark that Geometry is an excellent Logic. And it must be owned, that ...when from the distinct Contemplation and Comparison of Figures, their Properties are derived, by a perpetual well-connected chain of Consequences, the Objects being still kept in view, and the attention ever fixed upon them; there is acquired a habit of reasoning, close and exact and methodical: which habit strengthens and sharpens the Mind ...
Berkeley (1734), sec. 2, emphasis added
It should be noted that though Berkeley described a “presentist” conception of rigor in this remark, he did not generally subscribe to such a conception.
- 11.
By constancy of the subject-bearing parts of a proof, I mean constancy or identity of subjects throughout the subject-bearing parts of a proof (or, more exactly, throughout the series of judgments and inferences which together make up a proof).
In speaking of subjectivally continuous proof, I mean roughly proof in which the subjects of the subject-bearing parts of a proof are in some sense continuous with each other and with the overall subject of the proof, even though they may not be constant. Roughly speaking, continuity in this sense assumes that though the subjects of the subject-bearing parts of a proof may be distinct, the transitions from one to another are in some important way(s) conservative. No clearer formulation of these ideas is necessary for my purposes here.
- 12.
In mentioning the “course” of a proof here, I am assuming that proofs are characteristically divided, or at least divisible, into stages or steps. Nothing I propose here, though, depends on a particular working out of this idea.
- 13.
This suggests that Lambert may have seen properly rigorous proof as allowing not only inclusion of axioms among the legitimate ultimate premises of a proof, but inclusion of other propositions as well—specifically, propositions which were commonly recognized as having a basicness appropriate for use in proofs of the propositions being proved. This suggests a view of proof in which the basic qualification for premises is that they be appropriately more basic than the theorems they’re used to prove.
Lambert didn’t say in a precise way what he took the salience of such relative basicness to be. It seems sensible enough, though, to allow for the possibility that there be propositions which are axiom-like in certain respects (e.g. their relative evidentness, or their relative evidensory primitivity), but not in others (e.g. their deductive power, or their simplicity). It is also sensible enough to hold that the basic aim of proof is to justify the seemingly less basic by the seemingly more basic to the fullest extent feasible or practicable. On such a view of the aim of proof, a proof which used relatively more basic propositions to justify relatively less basic propositions could be seen as making progress even if the progress made were not that of justificative reduction to the most basic propositions.
- 14.
Lambert raised a related question as well, namely, whether, supposing the parallel postulate to not be so derivable, it might nonetheless become derivable by adding to the basic Euclidean propositions other propositions which have “the same evidentness” (die gleicher Evidenz hätten) (loc. cit.) as them (i.e. the basic Euclidean propositions). This, however, seems to have been more a comment concerning how to think about the independence of the parallel postulate and its significance than a comment concerning premisory rigor per se.
- 15.
There are indications that Lambert took Euclid to have been trying to develop a means of arguing which left no room for thought or judgment concerning things-in-themselves in geometrical reasoning. He saw the axioms as functioning symbolically, not semantically.
- 16.
Cf. Todhunter (1869).
- 17.
- 18.
On one variation, Pasch’s axiom states that, in a plane, if a line that does not pass through a vertex of a triangle intersects one side of it internally (i.e., at a point between vertices of the triangle), it then internally intersects another side and externally intersects the third.
- 19.
Here by the “use” of a geometrical figure I mean a justificative appeal to a judgment(s) concerning the properties of said diagram or of the figure(s) it may be taken to represent. The justification of such a judgment is presumably based on some type of “diagrammatic” grasp or examination of the figure involved.
- 20.
By a broadly logical understanding of this independence, I mean a view according to which to know that a proposition follows deductively from other propositions, it is not (broadly) logically necessary to know or to be in any way aware of senses, referents or images commonly associated with non-logical terms these propositions contain.
- 21.
This is a way of affirming the traditional idea that, properly speaking, deductive validity ought only to depend on the (logical) forms of the premises and conclusion of an inference.
- 22.
Here, by contentual use of an expression or figure \(\mathcal {E}\), I mean, roughly, use of a judgment \(\mathcal {J}\) to justify belief in the validity of an inference or proof where the (propositional) content of \(\mathcal {J}\) is in part determined by the content of \(\mathcal {E}\).
I am also supposing that, to count as being proper to a theory or subject-area \(\tau \), a use of \(\mathcal {E}\) must be thought to apply in some special or distinctive—some asymmetric—way, or to some asymmetric extent, to reasoning belonging to \(\tau \).
- 23.
Cf. “Mathematics is a system with two parts that should be distinguished. The first, properly mathematical, part, is focused exclusively on deduction. The second makes deduction possible by introducing and elucidating a series of insights that are to serve as material for deduction.”
Pasch (1918), 228
For a useful discussion of this and related ideas of Pasch’s see Pollard (2010).
- 24.
The text of this address was published as Hilbert (1928).
- 25.
According to this view, a proof is a finite sequence of judgments whose propositional contents are judged to stand in an appropriate deductive relationship to one another. This traditional view of proof, however, was something that Brouwer shared with many a non-intuitionist. Thus, though Hilbert directed his criticism towards Brouwer, it might just as justifiably have been aimed at Frege (cf. Frege 1906, 387), or any of a number of other thinkers of the late nineteenth and early twentieth centuries.
- 26.
- 27.
Hilbert referred to such processes of reasoning as “formaler Denkprozesse” in later writings (cf. Hilbert 1930, 380).
- 28.
To be more exact, the view described here was taken to apply to what Hilbert and Bernays later referred to as formal (formale) axiomatic reasoning, a type of axiomatic reasoning they distinguished (cf. Hilbert and Bernays 1934, §1) from contentual (inhaltliche) axiomatic reasoning. “[I]n contentual axiomatics (inhaltlichen Axiomatik)”, they said, “the basic relations are taken to be something found in experience or in intuitive conception (anschaulicher Vorstellung), and thus something contentually determined, about which the sentences of the theory make assertions (Behauptungen).” (Hilbert and Bernays 1934, 6.)
In formal axiomatization (formale Axiomatik), on the other hand, “the basic relations are not taken as having already been determined contentually. Rather, they are determined implicitly by the axioms from the very start. And in all thinking with an axiomatic theory only those basic relations are used that are expressly formulated in the axioms.” (op. cit., 7).
In his proof-theoretic writings Hilbert sometimes wrote ‘axiomatic’ where, more strictly speaking, he meant ‘formal axiomatic’.
- 29.
Cf. Hilbert (1926), 177 for one of a number a similar statements by Hilbert.
- 30.
I put ‘premises’ and ‘conclusions’ in scare-quotes because, in the present case, they are formulae, and not what premises and conclusions have traditionally been taken to be, namely, propositions or propositional attitude-takings (e.g., judgement or hypothesis).
- 31.
More specifically, what he and Bernays called formal axiomatic reasoning.
- 32.
Roughly speaking, exhibition in the current sense consists in the presentation (whatever, exactly, that might mean) of a particular concrete expression as an exemplar for other concrete expressions—specifically, expressions whose external features are sufficiently similar to those of the exemplar to qualify them as tokens of the same type as it.
- 33.
In Hilbert’s view, formal axiomatic thinking was not only the “basic instrument of all theoretical research” (Hilbert 1928, 4), it was also a general and pervasive form of human thought.
In our theoretical sciences we are accustomed to the use of formal thought processes (formaler Denkprozesse) and abstract methods ...[But] already in everyday life (täglichen Leben) one uses methods and concept-constructions (Begriffsbildungen) which require a high degree of abstraction and which only become plain through unconscious application of the axiomatic method (nur durch unbewußte Anwendung der axiomatischen Methoden verständlich sind). Examples include the general process of negation and, especially, the concept of infinity.
Hilbert (1930), 380
The last sentence of this remark raises questions concerning how Hilbert might have understood “unconscious” applications of the formal axiomatic method. Would it be possible to unconsciously apply a method of reasoning whose essence is consciousness of its own elements? As a matter of strict logical possibility, the answer would seem to be ‘yes.’ Whether this represents some other type of incoherence, though, is more difficult to say and something I lack space to consider further here.
- 34.
- 35.
Cf. Hilbert (1928), 1.
- 36.
By ‘expression’ here, I mean simply a string of characters in a language. I do not mean that this string serves to express a semantical content of some type.
- 37.
The need to bring syntactical categories into the picture is necessary in order to distinguish between similarly shaped syntactical objects that belong to different syntactical categories (e.g., a formula considered as a line of a proof versus a similarly shaped object which is taken to constitute a one line proof) .
- 38.
Hilary glossed the term “quantification theory” as “first-order logic” on p. 31 of Putnam (1984).
References
Berkeley, G. (1734). The analyst; or, a discourse addressed to an infidel mathematician. Wherein it is examined whether the object, principles, and inferences of the modern analysis are more distinctly conceived, or more evidently deduced, than religious mysteries and points of faith, Printed for J. Jonson, London.
Bour, P.-E., Rebuschi, M., & Rollet, L. (Eds.). (2010). Construction. London: College Publications.
Brouwer, L. E. J. (1923). Über die Bedeutung des Satzes vom ausgescholessenen Dritten in der Mathematik, insbesondere in der Funktiontheorie. Journal für die reine und angewandte Mathematik, 154, 1–7. English translation in van Heijenoort (1967), pp. 334–345. Page references are to this translation.
Brouwer, L. E. J. (1928). Intuitionistische Betrachtungen über den Formalismus. Koninklijke Akademie van wetenschappen de Amsterdam, Proceedings of the Section of Sciences, 31, 324–329. English translation with introduction in van Heijenoort (1967), pp. 490–492. Page references are to this translation.
Cellucci, C., Grosholz, E., & Ippoliti, E. (Eds.). (2011). Logic and knowledge. Newcastle upon Tyne: Cambridge Scholars Publishing.
Detlefsen, M. (2005). Formalism. In Shapiro (2005), pp. 236–317.
Detlefsen, M. (2010). Rigor, Re-proof and Bolzano’s Critical Program. In Bour et al. (2010), pp. 171–184.
Detlefsen, M. (2011). Dedekind against intuition: Rigor, scope and the motives of his logicism. In Cellucci et al. (2011), pp. 273–288.
Engel, F., & Stäckel, P. (1895). Die Theorie der Parallelinien von Euklid bis auf Gauss. Leipzig: Teubner.
Frege, G. (1906). Über die Grundlagen der Geometrie, Jahresbericht der Deutschen Mathematiker-Vereinigung, 15, 293–309, 377–403, 423–430. English translation in Frege (1971).
Frege, G. (1971). Gottlob Frege: On the foundations of geometry and formal theories of arithmetic, trans. and ed., with an introduction, by Eike-Henner W. Kluge. New Haven: Yale University Press.
Freudenthal, H. (1962). The main trends in the foundations of geometry in the 19th century. In Suppes et al. (1962), pp. 613–621.
Hilbert, D. (1899). Grundlagen der Geometrie, Published in Festschrift zur Feier der Enthüllung des Gauss-Weber Denkmals in Göttingen. Leipzig: Teubner.
Hilbert, D. (1900). Über den Zahlbegriff. Jahresbericht der Deutschen Mathematiker-Vereinigung, 8, 180–194.
Hilbert, D. (1922). Neubegründung der Mathematik. Erste Mitteilung. Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Universität, 1, 157–77. Page references are to the reprinting in Hilbert (1935), Vol. III.
Hilbert, D. (1926). Über das Unendliche. Mathematische Annalen, 95, 161–190.
Hilbert, D. (1928). Die Grundlagen der Mathematik. Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität, 6, 65–85. Reprinted in Hamburger Einzelschriften, 5, pp. 1–21. Leipzig: Teubner. Page references are to the latter.
Hilbert, D. (1930). Naturerkennen und Logik. Die Naturwissenschaften, 18, 959–963. Printed in Hilbert (1935), Vol. III. Page references are to this printing.
Hilbert, D., & Bernays, P. (1934). Grundlagen der Mathematik (Vol. I). Berlin: Springer. Page references are to Hilbert and Bernays (1968), the second edition of this text.
Hilbert, D. (1932–1935). Gesammelte Abhandlungen, three volumes. Bronx: Chelsea Pub. Co.
Hilbert, D., & Bernays, P. (1968). Grundlagen der Mathematik (Vol. I, 2nd ed.). Berlin: Springer.
Huntington, E. V. (1911). The fundamental propositions of algebra. Long Island University, Brooklyn: Galois Institute Press.
Lambert, J. H. (1786). Theory der Parallelinien I. Leipziger Magazin für reine und angewandte Mathematik, 1(2), 137–164. Reprinted in Engel and Stäckel (1995), pp. 152–176.
Lambert, J. H. (1786). Theory der Parallelinien II. Leipziger Magazin für reine und angewandte Mathematik, 1(3), 325–358. Reprinted in Engel and Stäckel (1995), pp. 176–207.
Pasch, M. (1882). Vorlesungen über neuere Geometrie. Leipzig: Teubner.
Pasch, M. (1918). Die Forderung der Entscheidbarkeit. Jahresbericht der Deutschen Mathematiker-Vereinigung, 27, 228–232.
Pollard, S. (2010). ‘As if’ reasoning in Vaihinger and Pasch. Erkenntnis, 73, 83–95.
Poncelet, J.-V. (1822). Traité des propriétés projectives de figures. Paris: Bachelier.
Putnam, H. (1984). Proof and experience. Proceedings of the American Philosophical Society, 128, 31–34.
Russell, B. (1902). The teaching of Euclid. The Mathematical Gazette, 2(33), 165–167.
Shapiro, S. (Ed.). (2005). Oxford handbook of the philosophy of mathematics and logic. Oxford: Oxford University Press.
Smith, C., & Bryant, S. (1901). Euclid’s elements of geometry: Books I–IV, VI and XI. London: Macmillan.
Suppes, P., Nagel, E., & Tarski, A. (Eds.) (1962). Logic, methodology, and philosophy of science: Proceedings of the 1960 International Congress. Stanford: Stanford U Press.
Todhunter, I. (1869). The elements of Euclid for the use of schools and colleges. London: Macmillan.
van Heijenoort, J. (1967). From Frege to Gödel: A source book in mathematical logic, 1879–1931. Cambridge: Harvard University Press.
Weyl, H. (1928). Diskussionsbemerkungen zu dem zweiten Hilbertschen Vortrag über die Grundlagen der Mathematik. Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität, 6, 86–88. Reprinted in Hamburger Einzelschriften, 5, pp. 22–24, Teubner, Leipzig, 1928. English translation in van Heijenoort (1967).
Weyl, H. (1944). David Hilbert and his mathematical work. Bulletin of the American Mathematical Society, 50, 612–654.
Whitehead, A. N. (1906). The axioms of projective geometry. Cambridge: University Press.
Young, J. W., Denton, W. W., & Mitchell, U. G. (1911). Lectures on fundamental concepts of algebra and geometry. New York: Macmillan Co.
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Detlefsen, M. (2018). Abstraction, Axiomatization and Rigor: Pasch and Hilbert. In: Hellman, G., Cook, R. (eds) Hilary Putnam on Logic and Mathematics. Outstanding Contributions to Logic, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-96274-0_11
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