§1. Dr. Paula Quinton has kindly shared with me the abstract of her recent work, titled „What „computing” means?” (see item 6 in References), and added some comments to more accurately explain her key ideas. Among the comments there was a link to S.Feferman’s paper „Turing’s ‚Oracle’. From Absolute to Relative computability and Back.” (2016).
In that paper I find a passage which seems to me crucial from an epistemological point of view, though it does not contain any direct references to philosophy. Before I quote and comment on it, let me express my personal attitude to some issues investigated by Ms. Quinton, and how I see their relation to the passage in question.
Frankly speaking, I am no expert in the issues of computing, and my interest in them is rather, so to say, old-fashioned, going back to Kazimierz Ajdukiewicz’s project in the early 1960s, concerned with the justification of statements and decisions. The subject was discussed at the very impressive international conference in 1961, chaired by Ajdukiewicz, and at a Polish conference, meant like a general rehearsal of Polish participants before that important international meeting.
What I remember best from those events, it is Andrzej Grzegorczyk discussion on the intuitive reasons of accepting axioms in mathematical theories. How such intuitions might be scientifcally justified? This issue seemed then like a puzzle which nobody was ready to solve in the situation (in philosophy of logic and mathematics) at that time. Let us call it
According to my knowledge, only Hao Wang’s publications in the nineteen-nineties, born from his penetrative discussions with Gödel, are likely to shed some light at the methodological problem of justifying axioms. They are related to the distinction between absolute and relative solvability as considered by Feferman.
Let the fact of adding a new axiom to a theory be exemplified by the transition from arithmetic without the Axiom of Complete Induction — ACI (like, e.g., Presburger arithmetic) to the theory enriched with this axiom. This results in an enormous amplification of the scope of solvability. However, such a success cannot be a decisive argument for the truth of so extended theory. One’s clear intuitive apprehension of the ACI’s truth, as witnessed by what Gödel says about his personal experience of its obviousness, also cannot suffice. However, there does exist an intersubjective evidence for ACI, to wit the gigantic range of its successful applications in every domain of human activity — astronomy,
accounting, engineering etc. — the success experienced during the tens of centuries in each civilization.
§2. Now it is in order to pay attention to Turing’s idea of oracle as presented in the mentioned (above) study by Feferman entitled: Turing’s ‚Oracle’: From Absolute to Relative computability and Back. The respective passage (slightly by me abbreviated) runs as follows (emphasis by Feferman).
>>The subject of effective computability began with what were offered as analyses of the absolute limits of effective computability, the immediate primary aim was to establish negative results of the effective unsolvability of various problems in logic and mathematics.
From this the subject turned to refined classifications of unsolvability. […] The germinal step, conceptually, was provided by Turing’s notion of computability relative to an ‚oracle’. At the hands of Post, this provided the beginning of the subject of degrees of unsolvability, less directly provided by Turing’s notion, but mplicit in it, were notions of uniform relative computability.<< Sec. 13.1, Introduction.
Turing did not say much about the way in which an oracle works, but made a very significant statement: that an oracle is no mechanical device, and that a machine equipped with that device can solve problems unsolvable for ordinary machines, lacking such a device. In other words, an oracle is able to find the value of an uncomputable function. More is said on the issue by Turing’s commentators like Andrew Hodges who offers convincing arguments that the decisions of an oracle are, in fact (in Turing’s intention) some acts of mathematical intuition. A similar stance has been taken by Roger Penrose. Thus, the amount of negative results diminishes with successive steps extending the scope of the solvability relative to an oracle. Now, as we know that the inventing of new, more powerful, systems of axioms is due to intellectual intuitions, and that axioms should be credited not only for their intutive obviousness, but much more for an immense number of successful applications, we attain at an answer to Ajdukiewicz-Grzegorczyk problem of the intuitive justification of axioms. This is to the effect that the intuition, being at the bottom of the process in question, proves its mettle by the fact the those axioms in which it has been expressed do successively pass the most severe exam of very differentiated and long-lasting applications.
§3. To conclude this part of discussion, let us observe that the interest in the idea of oracle, as involving the theory of grades of relative computability (solvability), is not confined to circle of speculative philosophers (to which belongs the present author). Presently it is found in the focus of highly technical discussions in mathematical logic. As well as in computer science; in the latter not only at the level of theoretical foundations, but also in the everyday practice. As testified by Feferman in the Conclusion of his penetrative study (Section 13.6.6):
>>The case had been made here that notions of relativized (as compared to absolute) computability theory are essentially involved in actual hardware and software design.<<
I made the above comment on some statements by Feferman, having been encouraged by the fact that his conribution was highly appreciated by Dr. Quinon. I am grateful for her message as when making use of it, I gained a new pespective in my thinking about computability.
In this perspective, I think that her research in the extensional equivalence and intensional differences between various treatments of Turingian model of computation should embrace the issue of the gradation of of relative computability; one may also say „relative solvability” for the former is a special case of the latter: computing is one of the ways of problem-solving.
Turings (1939) model of problem-solving makes more precise the notion of mathematical intuition, and then we can moce precisely put important questions, say: whether such an intuition can be obtained by artificial intelligence?
Now, when considering other theoretical models of computability (as those of Post, Church, Markov, Kleene, etc.), can we find in them an analog to relative computability as defined in terms of oracle? May be, the answer in the affirmative would be trivial, but I ask so being aware of my limited in that area expertise.
Another result which may be expected from Dr. Quinon’s inquiries is related to the question being the title of her abstract: „What „computing” means?” Should the notion of computing embrace the non-algorithmic activity performed by an oracle, namely that of finding the value (does it mean computing?) of an uncomputable function? If there is a puzzle in this question, what should be done to improve the terminology?
There would be more interesting issues inspired by Quinon’s project, but let those listed above suffice for the time being.
Items 1 and refer to the present author’s articles in which the concept of oracle is discussed, fairly extensively, with quoting and commenting Turing’s (1939) texts. Note that in both titles there appears the phrase „ever higher solvability” closely related to the notions of effective computability, effective unsolvability and relativized (as compared to absolute) computability — as employed by Feferman (quotations in §1 and §2 and the title in §1). Items 3 and 4 are concerned with the idea of oracle. Item 5 is related to that idea as dealing with Gödel’s theorem which can be used to exemplify Turing’s relative computability with respect to oracles.
1. „The progress of science from a computational point of view: the drive towards ever higher solvability” in: Foundations of Computing and Decision Sciences, Volume 44, Issue1 (2019), pp.11-26 — the journal published by the University of Technology in Poznań.
2. A.M. Turing. „Systems of Logic Based on Ordinals”. 1939. URL – local copy: turing-1939.pdf.
3. Robert I. Soare. Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory. 2009. URL – local copy: soare.pdf.
4. Samuel R. Buss. On Gödel’s Theorems on Lengths of Proofs I:
Number of Lines and Speedup for Arithmetics. 1992. URL – local copy : god-buss.pdf.
5. Paula Quinon. Abstract of the paper: What computiong” means
The Church-Turing thesis identifies the intuitive concept of what means ˙to compute˙ with a formal model of computation. There are multiple models of computation and all of them can be proved to be extensionally equivalent (they capture the same functions, such as ˙identity˙, or ˙the next element of the sequence˙). However, despite the extensional equivalence, models differ intensionally (they capture different aspects of computation, for instance computations on abstract natural numbers are intensionally different from computations performed by a machine using concrete electric signals). The main objective of this project is to characterize intensional differences between various concepts of computation.
When browsing my notes on solvability, I found a very illuminating remark by Gregory Chaitin. His phrase „intuition and creativity” may be interpreted as an exemplification of oracle, while the infinite chain of ever stronger axiomatic systems does resemble Turing’s ordering of logics.
Gödel’s own belief was that in spite of his incompleteness theorem there is in fact no limit to what mathematicians can achieve by using their intuition and creativity instead of depending only on logic and the axiomatic method. He believed that any important mathematical question could eventually be settled, if necessary by adding new fundamental principles to math, that is, new axioms or postulates. Note however that this implies that the concept of mathematical truth becomes something dynamic that evolves, that changes with time, as opposed to the traditional view that mathematical truth is static and eternal. See „Chaitin interview for Simply Gödel website” (9 February 2008).
Consider the phrase I have emphasised (through red type). Does the process of „adding new fundamental principles” belong to the processes of computation? If so, which model of computation would explain the intuitive meaning of the a phrase like that? This is a question which seems to be implied by Quinin’s plan of comparing the existing explications of the term „computing”.