Abstract
It is frequently claimed (see e.g. Rav, A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices. Philosophia Mathematica (III) 15:291–320, 2007) that the formalization of a mathematical proof requires a quality of understanding that subsumes all acts necessary for checking the proof and that, consequently, automatic proof checking cannot lead to an epistemic gain about a proof. We present a project developing what is sometimes called a ‘fortified formalism’ and argue, taking a phenomenological look at proof understanding, that proofs can be (and often are) given in a way that allows a formalization sufficient for producing an automatically checkable write-up, but does not subsume checking.
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Change history
29 January 2020
The abstract of this chapter was initially published with error. The chapter has been updated with the corrected abstract as given below.
Notes
- 1.
The reasoning here is roughly this: If ϕ is not formally derivable from S, then there is a model of S + ¬ϕ, i.e. a way to interprete the occurring notions in such a way that S becomes true, but ϕ becomes false. In the presence of such an interpretation, no argument claiming to deduce ϕ from S can be conclusive. A similar point is made by Kreisel in Kreisel (1965).
- 2.
For example, Boolos’ proof has recently been formalized, see Benzmüller and Brown (2007). The matter appears to be more a question of the choice of the formal system than one of formal vs. informal proof or of first-order vs. higher-order logic: When a proof is ‘possible, but far too long’ in a certain first-order axiomatic system, there are usually natural stronger systems that allow for the necessary abbreviations.
- 3.
Cf. Rav (2007, p. 301).
- 4.
Rav (2007, p. 306).
- 5.
- 6.
Another related perspective is that of Martin-Löf given in Martin-Löf (1980) (in particular on p. 7): He distinguishes ‘canonical’ from ‘noncanonical’ or ‘indirect’ proofs, where a ‘noncanonical proof’ is a ‘method’ or a ‘program’ for producing a ‘canonical’ proof; as an example, an indirect proof that 1235 + 5123 = 5123 + 1235 would consist in first proving the general law of commutativity for addition and then instantiating it accordingly rather than carrying out the constructions described by both sides of the equation and checking that they actually lead to the same result. In a similar way, we may view an informal high-level argument as a recipe for obtaining a proof in which every formerly implicit inferential step is actually carried out. Of course, we don’t need to go along with Martin-Löf’s constructivist approach concerning mathematics here: The checking of non-constructivistic proofs is – regardless of how one views their epistemological value – a cognitive act which is, along with the underlying notion of correctness and its relation to automatization, accessible to a phenomenological analysis. For a perspective on the fulfillment of mathematical intentions based on the notion of construction, see van Atten (2010) and van Atten and Kennedy (2003).
- 7.
E.g. the statement ‘The theory of relativity is green’ is arguably neither true nor false, which nevertheless doesn’t contradict the principle of the excluded middle.
- 8.
See Lohmar (1989) for a further discussion of this point.
- 9.
Making a concept distinct, i.e. making distinct that which is meant by the word by itself, is a procedure which takes place within the sphere of pure thinking. Before the least step of clarification is performed, while no or a completely inadequate intuition is associated with the word, we can reflect upon that which is meant, i.e. in ‘decahedron’: a solid, a polyhedron, regular, with ten congruent faces. (…) With a clarification, we transcend the sphere of mere meanings and thinking concerning meanings, we match the meanings with the noematic content of intuition. [Translation by the author]
- 10.
Indeed, as a working mathematician, one occasionally experiences the perception of a vague proof idea which seems quite plausible until one attempts to actually write it down. When one finally does, it becomes apparent that the argument has serious structural issues, e.g. being circular. This particularly happens when one deals with arguments and definitions using involved recursions or inductions.
- 11.
Singh (2000), appendix, or see e.g. http://www.quickanddirtytips.com/education/math/how-to-prove-that-1-2
- 12.
This point is also discussed in Martin-Löf (1987), p. 418: ‘And it is because of the fact that we make mistakes that the notion of validity of a proof is necessary: If proofs were always right, then of course the very notion of rightness or rectitude would not be needed.’
- 13.
We thank Marcos Cramer for the kind permission to use the Naproche version of this text in our paper.
- 14.
Historically, at least. The deductive style dominant in mathematical textbooks confronts the student with the inverse problem: Namely making sense of a seemingly unmotivated given formal definition.
- 15.
An excellent example of the delicate dialectics involved in forming definitions of intuitive concepts is the notion of polyhedron in Euler’s polyhedron formula as discussed in Landau (2004). Sometimes, this is the really hard part in creating new mathematics. Another prominent example is the way how the intuitive notion of computability was formalized by the concept of the Turing machine.
- 16.
Suppose, for example, that someone came up with a non-recursive function that one can evaluate without investing original thought so that one is inclined to accept the evaluation of this function as an instance of ‘calculation’, thus disproving the Church-Turing thesis. As a consequence, recursiveness would lose its status as an exact formulation of the intuitive concept of calculation. But this would not affect the correctness of recursion theory.
- 17.
This formulation is sometimes disputed as not correctly capturing the argument Cauchy had in mind. Some claim that Cauchy meant the variables implicit in his text to not only range over what is now known as the set of reals, but also over infinitesimals. However, the formulation we offer captures the way the proof was and still is understood and at first sight considered correct by many readers, so we will not pursue this historical question further.
- 18.
Even more, taking the speculation a bit further, being able to meaningfully and convincingly participate in this kind of discourse is a plausible criterion for not calling something a ‘machine’ any more. (Moreover, the definiteness of a machine’s response, which is a main motivation for striving for automatization in the first place, is lost when a machine becomes merely another participant of a discourse.) Of course, the rules of a discourse are made by its participants; so in the end, the possibility of computers becoming influential even in the conceptual part of mathematics might boil down to the question whether we are willing to accept a computer as a participant in such a debate on equal terms. However, we now have reached a level of speculation at which it is better to stop.
- 19.
There are, however, several phenomenological accounts of mathematical proofs focusing on other aspects. We are thankful to the anonymous referee of an earlier version of this paper for pointing out to us four very worthwhile works on the phenomenology of proofs, and we take the opportunity to briefly discuss their relation to our work:
In Tragesser (1992), R. Tragesser points out that, in the philosophy of mathematics, actual proof practice is often ignored in favour of or replaced with formal proofs and remarks that this ‘idealizes away’ many elements that are important for mathematics. To remedy this, he proposes phenomenology as a better approach, where phenomenology means focusing on ‘the thing itself’ and striving for a deep an understanding as possible rather than being satisfied with pragmatical considerations. In particular, he emphasizes that possible proofs are as good as actual proofs and that this is a sense in which mathematics can be said to be a priori. His approach does not build on Husserl’s phenomology and also does not apply, as ours does, to the analysis of mistakes, as for Tragesser, a proof is always a correct proof.
A similar critique of focusing on formal representations of proofs is given by G.-C. Rota in Rota (1997), who also presents a similar understanding of phenomenological analysis, which should consider as close as possible actual proofs without ignoring important aspects as belonging to other fields – such as psychology – or due to confusing description with norms. Rota in particular points out that ‘proof’ has aspects beyond ‘verification’, such as understanding, degrees of understanding, giving ‘reasons’, different kinds of proof and the heuristical value that proofs and theorems play for the further development of mathematics. He also gives historical examples to substantiate his thesis that mathematics is really a strive for understanding in which theorems and proofs both play equal roles.
In Sundholm (1993), G. Sundholm notes that there is a ‘tension’ between various plausible claims put forward concerning proofs, which leads him to distinguish between proof-acts (series of mental acts), proof objects (the result obtained at the end of a proof) and proof-traces (the written presentation, that allows others to reproduce the series of acts). His work does not build on Husserl’s analysis of logic, but is related to our work as the present paper can be viewed as a closer analysis of the process of reproducing a proof-act on the basis of a proof-trace, distinguishing different qualities that such an reproduction can have and attempting to determine which of these needs to be achieved in order to check the proof for correctness or to produce an automatically verifiable version.
In Hartimo and Okada (2016), M. Hartimo and M. Okada compare Hankel’s approach to formal arithmetic and mathematics to Husserl’s and demonstrate strong parallels between Husserl’s treatment of arithmetic and term-rewriting systems commonplace in computer science. As an example given on p. 955 of Hartimo and Okada (2016) that goes back to Husserl, an arithmetical expression like 18+48 is here considered to be ‘fulfilled’ by a normal form expression, in this case 66, and the possibility of fulfilling the intentions signified by such expressions is demonstrated by showing how to reduce them to such a unique normal form by a certain algorithm.
- 20.
To appreciate this difference, one might consider the equation ((a + b) + c) + d = ((d + c) + b) + a over the reals, which is obviously true for a human reader who thinks of addition as taking the union of two quantities. In our experience with number-theoretical texts in Naproche (Carl), the automatic prover, having to derive this from commutativity and associativity of addition, often got lost in the countless alternative possibilities which rule to apply. This is a striking example for the pragmatical difference between formal definitions and intuitive concepts.
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Acknowledgements
We thank Marcos Cramer for the kind permission to use his Naproche version of Rav’s proof Sect. 14.4.1. We thank Dominik Klein and an anonymous referee for various valuable comments on former versions of this work that led to considerable improvements. We also thank Heike Carl for her thorough proofreading.
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Carl, M. (2019). Formal and Natural Proof: A Phenomenological Approach. In: Centrone, S., Kant, D., Sarikaya, D. (eds) Reflections on the Foundations of Mathematics. Synthese Library, vol 407. Springer, Cham. https://doi.org/10.1007/978-3-030-15655-8_14
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